TIGHT BINDING BOOK Page No 104 missing CO > OU 164019 THEORY AND APPLICATION OF INFINITE SERIES BLACKIE & SON LIMITED 16/18 William IV Street, Charing Crosi,, LONDON VV C 2 17 Stanhope Street, GLASGOW BLACKJE & SON (INDIA) LIM1TKD 103/5 Fort Street, BOMBAY BLACKIH & SON (CANADA) LIMITED TORONTO THEORY AND APPLICATION OF INFINITE SERIES BY DR. KONRAD KNOPP PROFESSOR OF MATHTCMATICb AT THE UNIVLRSITY OF TUBINGEN Translated from the Second German Edition and revised in accordance with the Fourth by Miss R. C. H. Young, Ph.D., L.esSc. BLACKIE & SON LIMITED LONDON AND GLASGOW First issued i o a8 Reprinted 1044, 1946 Second Publish Fdition, translated ft o, the Fourth German Edition, 1951 Rfpntitrd TO 54 Printed in Great Britain by Blackie & Son, Ltd., Glasgow From the preface to the first (German) edition. There is no general agreement as to where an account of the theory of infinite series should begin, what its main outlines should be, or what it should include. On the one hand, the whole of higher analysis may be regarded as a field for the application of this theory, for all limiting processes including differentiation and integration are based on the investigation of infinite sequences or of infinite series. On the other hand, in the strictest (and therefore narrowest) sense, the only matters that arc in place in a textbook on infinite series are their definition, the manipulation of the symbolism connected with them, and the theory of convergence. In his "Vorlesungen uber Zahlcn- und Funktioncnlehre", Vol. 1, Part 2, A. Pringsheim has treated the subject with these limitations. There was no question of offering anything similar in the present book. My aim was quite different: namely, to give a comprehensive account of all the investigations of higher analysis in which infinite series are the chief object of interest, the treatment to be as free from assump- tions as possible and to start at the very beginning and lead on to the extensive frontiers of present-day research. To set all this forth in as interesting and intelligible a way as possible, but of course without in the least abandoning exactness, with the object of providing the student with a convenient introduction to the subject and of giving him an idea of its rich and fascinating variety such was my vision. The material grew in my hands, however, and resisted my efforts to put it into shape. In order to make a convenient and useful book, the field had to be restricted. But I was guided throughout by the ex- perience I have gained in teaching I have covered the whole of the ground several times in the general course of my work and in lectures at the universities of Berlin and Konigsbcrg and also by the aim of the book. It was to give a thorough and reliable treatment which would be of assistance to the student attending lectures and which would at the same time be adapted for private study. The latter aim was particularly dear to me, and this accounts for the form in which I have presented the subject-matter. Since it is gener- ally easier especially for beginners to prove a deduction in pure mathematics than to recognize the restrictions to which the train of reasoning is subject, I have always dwelt on theoretical difficulties, and VI Preface. have tried to remove them by means of repeated illustrations; and although I have thereby deprived myself of a good deal of space for important matter, I hope to win the gratitude of the student. I considered that an introduction to the theory of real numbers was indispensable as a beginning, in order that the first facts relating to convergence might have a firm foundation. To this introduction I have added a fairly extensive account of the theory of sequences, and, finally, the actual theory of infinite series. The latter is then constructed in two storeys, so to speak: a ground-floor, in which the classical part of the theory (up to about the stage of Cauchy's Analyse algebrique) is expounded, though with the help of very limited resources, and a super- structure, in which I have attempted to give an account of the later developments of the 19 th century. For the reasons mentioned above, I have had to omit many parts of the subject to which I would gladly have given a place for their own sake. Semi-convergent series, Euler's summation formula, a detailed treatment of the Gamma-function, problems arising from the hypjr- geometric series, the theory of double series, the newer work on power series, and, in particular, a more thorough development of the last chapter, that on divergent scries all these I was reluctantly obliged to set aside. On the other hand, I considered that it was essential to deal with sequences and series of complex terms. As the theory runs almost parallel with that for real variables, however, I have, from the beginning, for- mulated all the definitions and proved all the theorems concerned in such a way that they remain valid without alteration, whether the "arbi- trary" numbers involved are real or complex. These definitions and theorems are further distinguished by the sign . In choosing the examples in this respect, however, I lay no claim to originality; on the contrary, in collecting them I have made extensive use of the literature I have taken pains to put practical applications in the fore-front and to leave mere playing with theoretical niceties alone. Hence there are e. g. a particularly large number of exer- cises on Chapter VIII and only very few on Chapter IX. Unfortunately there was no room for solutions or even for hints for the solution of the examples. A list of the most important papers, comprehensive accounts, and textbooks on infinite series is given at the end of the book, immediately in front of the index. Kdnigsberg, September 1921. Preface. VII From the preface to the second (German) edition. The fact that a second edition was called for after such a remarkably short time could be taken to mean that the first had on the whole been on the right lines. Hence the general plan has not been altered, but it has been improved in the details of expression and demonstration on almost every page. The last chapter, that dealing with divergent series, has been wholly rewritten, with important extensions, so that it now in some measure provides an introduction to the theory and gives an idea of modern work on the subject. Kdnigsberg, December 1923. Preface to the third (German) edition. The main difference between the third and second editions is that it has become possible to add a new chapter on Euler's summation formula and asymptotic expansions, which I had reluctantly omitted from the first two editions. This important chapter had meanwhile appeared in a similar form in the English translation published by Blackie & Son Limited, London and Glasgow, in 1928. In addition, the whole of the book has again been carefully revised, and the proofs have been improved or simplified in accordance with the progress of mathematical knowledge or teaching experience. This applies especially to theorems 269 and 287. Dr. W, Schobe and Herr P. Securius have given me valuable assist- ance in correcting the proofs, for which I thank them heartily. Tubingen, March 1931. Preface to the fourth (German) edition. In view of present difficulties no large changes have been made for the fourth edition, but the book has again been revised and numerous details have been improved, discrepancies removed, and several proofs simplified. The references to the literature have been brought up to date. Tubingen, July 1947. VIII Preface. Preface to the first English edition. This translation of the second German edition has been very skil- fully prepared by Miss R. C. //. Young, L. es Sc. (Lausanne), Research Student, Girton College, Cambridge. The publishers, Messrs. Blackie and Son, Ltd., Glasgow, have carefully superintended the printing. In addition, the publishers were kind enough to ask me to add a chapter on Enter's summation formula and asymptotic expansions. I agreed to do so all the more gladly because, as I mentioned in the original pre- face, it was only with great reluctance that I omitted this part of the sub- ject in the German edition. This chapter has been translated by Miss W. M. Deans, B.Sc. (Aberdeen), M.A. (Cantab.), with equal skill. I wish to take this opportunity of thanking the translators and the publishers for the trouble and care they have taken. If as I hope my book meets with a favourable reception and is found useful by English- speaking students of Mathematics, the credit will largely be theirs. Tubingen, February 1928. Konrad Knopp. Preface to the second English edition. The second English edition has been produced to correspond to the fourth German edition (194/7). Although most of the changes are individually small, they have none- theless involved a considerable number of alterations, about half of the work having been re-set. The translation has been carried out by Dr. R. C. H. Young who was responsible for the original work. Contents. Page Introduction ., 1 Part I. Real numbers and sequences. Chapter 1. Principles of the theory of real numbers. 1. The system of rational numbers and its gaps 3 2 Sequences of rational numbers 14 3 Irrational numbers 23 4. Completeness and uniqueness of the system of real numbers . . . Jj3 5. Radix fractions and the Ded'-kind section 37 Exercises on Chapter 1 (1 8) 42 Chapter II. Sequences of real numbers. 6. Arbitrary sequences and arbitrary null sequences 43 7. Powers, roots, and logarithms Special null sequences 49 8. Convergent sequences 64 $ The two main criteria ... ... . 78 10 Limiting 1 points and upper and lower limits 89 11. Infinite series, infinite products, and infinite continued fractions . 98 Exercises on Chapter II (933) . . . . 106 Part II. Foundations of the theory of infinite series. / Chapter 111. * Series of positive terms. 12. The first principal criterion and the two comparison tests .... 110 13. The root test and the ratio test 116 14 Series of positive, monotone decreasing terms 120 Exercises on Chapter III (3444) 125 ** (CJ51) X Contents. Chapter IV. Page Series of arbitrary terms. 15. The second principal criterion and the algebra of convergent series 126 16. Absolute convergence. Derangement of series 136 17. Multiplication of infinite series 146 Exercises on Chapter IV (4503) 149 / Chapter V. Power series. 18. The radius of convergence 151 19. Functions of a real variable . . 158 20. Principal piopertics of functions leprcsented by power series . . . 171 21. The algebra of power series 179 Exercises on Chapter V (64 -73) 188 Chapter VI The expansions of the so-called elementary functions. 22. The rational functions 189 23. The exponential function 191 24. The trigonometrical functions 198 25. The binomial series 208 26. The logarithmic series 211 27. The cyclometrical functions 213 Exercises on Chapter VI (74 -84) 215 Chapter VII. Infinite products. 28. Products with positive terms 218 29. Products with arbitrary terms. Absolute convergence ... . 221 30. Connection between series and products. Conditional and unconditional convergence . 226 Exercises on Chapter Vll (8599) 228 Chapter VIII. s^ Closed and numerical expressions for the sums of series. 31. Statement of the problem -J30 32. Evaluation of the sum of a series by means of a closed expression 232 33. Transformation of series 240 34. Numerical evaluations 247 35. Applications of the transformation of series to numerical evaluations 260 Exercises on Chapter VIIF (100132) 267 Contents. Part III Development of the theory. y Chapter IX. Series of positive terms. 36. Detailed study of the two comparison tests . 274 37. The logarithmic scales . . 278 38. Special comparison tests of the second kind 284 39. Theorems of Abel, Dint, and Prin^heim t and their application to a fresh deduction of the logarithmic scale of comparison tests . . . 290 40. Series of monotonely diminishing positive terms 294 41. General remarks on the theory of the convergence and divergence of series of positive terms 298 42. Sy sterna ti/ation of the general theory of convergence 305 Exercises on Chapter IX (138141) 311 Chapter X. \/ Series of arbitrary terms. \ 43. Tests of convergence for series ot arbitrary terms 312 I 44. Rearrangement of conditionally convergent series 318 | 45. Multiplication of conditionally convergent series 320 Exercises on Chapter X (142153; 324 , Chapter XI. Series of variable terms (Sequences of functions). 46. Uniform convergence 326 47. Passage to the limit term by term 338 48 Tests of uniform convergence 344 49. Fourier scries 350 A. Euler's formulae 350 B. Dinchlet's integral 356 C. Conditions of convergence 364 50. Applications of the theory of Fourier series 372 51. Products with variable terms 380 Exercises on Chapter XI (154 173J 385 Chapter XII. Series of complex terms. 52. Complex numbers and sequences 388 53. Series of complex terras 396 54. Power series. Analytic functions 401 XII Contents. Page. 55. The elementary analytic functions 410 I. Rational functions 410 II. The exponential function 411 III. The functions cosz and sin z 414 IV. The functions cot z and tan* 417 V. The logarithmic scries 419 VI. The inverse sine series 4*21 VII. The inverse tang-cnt series 422 VIII. The binomial series 423 56. Series of variable terms. Uniform convergence. Weierstrass* theo- rem on double series *- 428 57. Products with complex terms 434 58. Special classes of series of analytic functions 441 A. Dinchlet's series 441 B. Faculty series 446 C. Lambert's series . 448 Exercises on Chapter XII (174199) 452 Chapter XI11. Divergent series. 59. General remarks on divergent series and the processes of limitation 457 60. The C- and H- processes 478 61. Application of C t - summation to the theory of Fourier series . . . 492 62. The A- process 498 63. The E- process 507 Exercises on Chapter XIII (200216) 516 Chapter XIV. Euler's summation formula and asymptotic expansions. 64. Euler's summation formula . 518 A. The summation formula 518 B. Applications 525 C. The evaluation of renviinders 531 65. Asymptotic scries 535 66. Special cases of asymptotic expansions 543 A. Examples of the expansion problem 543 B. Examples of the summation problem 548 Exercises on Chapter XIV (217-225) 553 Bibliography 556 Name and subject index ........ 557 Introduction. The foundation on which the structure of higher analysis rests is the theory of real numbers. Any strict treatment of the foundations of the differential and integral calculus and of related subjects must in- evitably start from there; and the same is true even for e. g. the cal- culation of roots and logarithms. The theory of real numbers first creates the material on which Arithmetic and Analysis can subsequently build, and with which they deal almost exclusively. The necessity for this has not always been realized. The great creators of the infinitesimal calculus Leibniz and Newton l and the no less famous men who developed it, of whom Eider 2 is the chief, were too intoxicated by the mighty stream of learning springing from the newly-discovered sources to feel obliged to criticize fundamentals. To them the results of the new methods were sufficient evidence for the security of their foundations. It was only when the stream began to ebb that critical analysis ventured to examine the fundamental con- ceptions. About the end of the 18 th century such efforts became stronger and stronger, chiefly owing to the powerful influence of Gauss 3 . Nearly a century had to pass, however, before the most essential matters could be considered thoroughly cleared up. Nowadays rigour in connection with the underlying number concept is the most important requirement in the treatment of any mathematical subject. Ever since the later decades of the past century the last word on the matter has been uttered, so to speak, by Weierstrass 4 in the sixties, and by Cantor 5 and Dedekind 6 in 1872. No lecture or treatise 1 Gottfried Wilhelm Leibniz, born in Leipzig in 1646, died in Hanover in 1716. Isaac Neivton, born at Woolsthorpe in 1642, died in London in 1727. Each discovered the foundations of the infinitesimal calculus independently of the other. 2 Leonhard Eider, born in Basle in 1707, died in St. Petersburg in 1783. 3 Karl Friedrich Gauss, born at Brunswick in 1777, died at Gottingen in 1853. 4 Karl Weierstrass, born at Ostenfelde in 1815, died in Berlin in 1897. The first rigorous account of the theory of real numbers which Weierstrass had expounded in his lectures since 1860 was given by G. Mittag-Leffler, one of his pupils, in his essay: Die Zahl, Einleitung zur Theone der analytischen Funktionen, The Tohoku Mathematical Journal, Vol. 17, pp. 157209. 1920. 5 Georg Cantor, born in St. Petersburg in 1845, died at Halle in 1918: cf. Mathem. Annalen, Vol. 5, p. 123. 1872. 6 Richard Dedekind, born at Brunswick in 1831, died there in 1916: cf. his book: Stetigkeit und irrationaJe Zahlen, Brunsuick 1872. 1 2 Introduction. dealing with the fundamental parts of higher analysis can claim validity unless it takes the refined concept of the real number as its starting- point. Hence the theory of real numbers has been stated so often and in so many different ways since that time that it might seem superfluous to give another very detailed exposition 7 : for in this book (at least in the later chapters) we wish to address ourselves only to those already acquainted with the elements of the differential and integral calculus. Yet it would scarcely suffice merely to point to accounts given elsewhere. For a theory of infinite series, as will be sufficiently clear from later developments, would be up in the clouds throughout, if it were not firmly based on the system of real numbers, the only possible foundation. On account of this, and in order to leave not the slightest uncertainty as to the hypotheses on which we shill build, we shall discuss in the following pages those idsas and data from the theory of real numbers which we shall need further on. We have no intention, however, of con- structing a statement of the theory compressed into smaller space but otherwise complete. We merely wish to make the main ideas, the most important questions, and the answers to them, as clear and prominent as possible. So far as the latter are concerned, our treatment throughout will certainly be detailed and without omissions; it is only in the cases of details of subsidiary importance, and of questions as to the complete- ness and uniqueness of the system of real numbers which lie outside the plan of this book, that we shall content ourselves with shorter indications. 7 An account which is easy to follow and which includes all the essentials is given by H. v. Mangoldt, Einfuhrung in die hohere Mathematik, Vol. I, 8 th edition (by K. Knopp), Leipzig 1944. The treatment of G. Kozvalezvski, Grundziige der Differential- und Integralrechnung, 6 th edition, Leipzig 1929, is accurate and concise. A rigorous construction of the system of real numbers, which goes into the minutest details, is to be found in A. Loezvy, Lehrbuch der Algebra, Part I, Leipzig 1915, in A. Pnngsheim, Vorlesungen uber Zahlen- und Funktionenlehre, Vol. I, Part I, 2 n(1 edition, Leipzig 1923 (cf. also the review of the latter work by H. Hahn, Gott. gel. Anzeigen 1919, pp. 321 47), and in a book by E. Landau exclusively devoted to this purpose, Grundlagen der Analysis (Das Rechnen mit ganzen, rationalen, irrationalen, komplexen Zahlen), Leipzig 1930. A critical account of the whole problem is to be found in the article by F. Bachmann, Aufbau des Zahlensystems, in the Enzyklopadie d. math. Wissensch., Vol. I, 2 nii edition, Part I, article 3, Leipzig and Berlin 1938. Part I. Real numbers and sequences. Chapter I. Principles of the theory of real numbers. 1. The system of rational numbers and its gaps. What do we mean by saying that a particular number is "known" or "given" or may be "calculated"? What does one mean by saying that he knows the value of 1/2 or n>> or lnat ne can calculate 1/5? A question like this is easier to ask than to answer. Were I to say that \/2 = l-414, I should obviously be wrong, since, on multi- plying out, 1-414 X 1-414 does not give 2. If I assert, with greater caution, that 1/2 = 1-4 142 135 and so on, even that is no tenable answer, and indeed in the first instance it is entirely meaningless. The question is, after all, how we are to go on, and this, without further indication, we cannot tell. Nor is the position improved by carrying the decimal further, even to hundreds of places. In this sense it may well be said that no one has ever beheld the whole of "V/2, not held it completely in his own hands, so to speak whilst a statement that 1/9 = 3 or that 35-7-7 = 5 has a finished and thorough- ly satisfactory appearance. The position is no better as regards the number n, or a logarithm or sine or cosine from the tables. Yet we feel certain that 1/2 and n and log 5 really do have quite definite values, and even that we actually know these values. But a clear notion of what these impressions exactly amount to or imply we do not as yet possess. Let us endeavour to form such an idea. Having raised doubts as to the justification for such statements as "I know 1/2", we must, to be consistent, proceed to examine how far one is justified even in asserting that he knows the number ^ or is given (for some specific calculation) the number ~. Nay more, the significance of such statements as "I know the number 97" or "for such and such a calculation I am given a = 2 and 6 = 5" would 4 Chapter I. Principles of the theory of real numbers. require scrutiny. We should have to enquire into the whole significance or concept of the natural numbers 1, 2, 3, ... This last question, however, strikes us at once as distinctly trans- gressing the bounds of Mathematics and as belonging to an order of ideas quite apart from that which we propose to develop here. No science rests entirely within itself: each borrows the strength of its ultimate foundations from strata above or below it, such as experi- ence, or theory of knowledge, or logic, or metaphysics, . . . Every science must accept something as simply given, and on that it may proceed to build. In this sense neither mathematics nor any other science starts without assumptions. The only question which has to be settled by a criticism of the foundation and logical structure of any science is what shall be assumed as in this sense "given"; or better, what minimum of initial assumptions will suffice, to serve as a basis for the subsequent development of all the rest. For the problem we are dealing with, that of constructing the system of real numbers, these preliminary investigations are tedious and trouble- some, and have actually, it must be confessed, not yet reached any entirely satisfactory conclusion at all. A discussion adequate to the present position of the subject would consequently take us far beyond the limits of the work w r e are contemplating. Instead, therefore, of shouldering an obligation to assume as basis only a minimum of hypotheses, we propose to regard at once as known (or "given", or "secured") a group of data whose deducibility from a smaller body of assumptions is familiar to everyone namely, the system of rational numbers, i. e. of numbers integral and fractional, positive and negative, including zero. Speaking broadly, it is a matter of common knowledge how this system may be constructed, if as a smaller body of assumptions only the ordered sequence of natural numbers 1, 2, 3, . . . , and their combinations by addition and multiplication, are regarded as "given". For everyone knows and we merely indicate it in passing how fractional numbers arise from the need of inverting the process of multiplication, negative numbers and zero from that of inverting the process of addition 1 . The totality, or aggregate, of numbers thus obtained is called the system (or set) of rational numbers. Each of these can be completely and literally "given" or "written down" or "made known" with the help of at most two natural numbers, a dividing bar and possibly a minus sign. For brevity, we represent them by small italic characters; #,&,..., x, y, . . . The following are the essential properties of this system: 1 See the works of Loewy, Pringsheim, and Landau mentioned in the Intro- duction; also O. Holder, Die Anthmetik in strenger Begrundung, 2" J edition, Berlin 1929; and O. Stolz and J. A. Gmeiner, Theoretische Arithmetik, 3 r<l edition, Leipzig 1911. 1. The system of rational numbers and its gaps. 5 1. Rational numbers form an ordered aggregate; meaning that between any two, say a and 6, one and only one of the three relations a < b. a = b, a > b necessarily holds 2 ; and these relations of "order" between rational numbers are subject to a set of quite simple laws, which we assume known, the only essential ones for our purposes being the Fundamental Laws of Order. 1. Invariably 3 a a. 2. a b always implies b - a. 3. a = b y b c implies a = c. 4. a ^ b y b < c, or a < b, b < c y implies 4 a < c. 2. Any two rational numbers may be combined in four distinct ways, referred to respectively as the four processes (or basic operations) of Addition, Subtraction, Multiplication, and Division. These operations can always be carried out to one definite result, with the single exception of division by 0, which is undefined and should be regarded as an entirely impossible or meaningless process; the four processes also obey a number of simple laws, the so-called Fundamental Laws of Arithmetic, and further rules cleducible therefrom. These too we shall regard as known, and state, concisely, those Fundamental Laws or Axioms of Arithmetic from which all the others may be inferred, by purely formal rules (i. e. by the laws of pure logic). I. Addition. 1. Every pair of numbers a and b has invariably associ- ated with it a third, c, called their sum and denoted by a + b. 2. a = a', b b' always implv a \ b -- a' + b'. 3. Invariably, a + b b + (Commutative Law). 4. Invariably, (a + b) + c = a + (b + c) (Associative Law). 5. a < b always implies a + c < b + c (Law of Monotony). II. Subtraction. To every pair of numbers a and b there corresponds a third number c, such that a + c b. 8 a > b and b < a are merely two different expressions of the same relation. Strictly speaking, the one symbol "<" would therefore suffice. 3 With regard to this seemingly trivial "law" cf. footnote 11, p. 9, remark 1 , p. 28, and footnote 24, p. 29. 4 To express that one of the relations of order: a < b, a 6, or a > b, does not hold, we write, respectively, a^b ("greater than or equal to", "at least equal to", "not less than"), a -t= b ("unequal to", "different from") or a *- 6. Kach of these statements (negations) definitely excludes one of the three relations and leaves undecided which of the other two holds good. 6 Chapter I. Principles of the theory of real numbers. III. Multiplication. 1. To every pair of numbers a and b there corresponds a third number c, called their product and denoted by a b. 2. a a', b b' always implies a b = a' b'. 3. In all cases ab = ba (Commutative Law). 4. In all cases (ab) c =-- a (b c) (Associative Law). 5. In all cases (a + b) c a c + b c (Distributive Law). 6. a < b implies, provided c is positive, a c <.b c (Law of Mono- tony). IV. Division. To every pair of numbers a and b of which the first is not there corresponds a third number c, such that a c = b. As already remarked, all the known rules of arithmetic, and hence ultimately all mathematical results, are deduced from these few laws, with the help of the laws of pure logic alone. Among these laws, one is distinguished by its primarily mathematical character, namely the V. Law of Induction, which may be reckoned among the fundamental laws of arithmetic and is normally stated as follows: If a set S 3)t of natural numbers includes the number 1, and if, every time a certain natural number n and all those less than n can be taken to belong to the aggregate, the number (n h 1) rniy be inferred also to belong to it, then $)J includes all the natural numbers. This law of induction itself follows quite easily from the following theorem, which appears even more obvious and is therefore normally called the fundamental law of the natural numbers : Law of the Natural Numbers. In every set of natural numbers that is not "empty" there is always a number less than all the rest. For if, according to the hypotheses of the Induction Law, we con- sider the set 9i of natural numbers not belonging to $)?, this set W must be "empty", that is, $ft must contain all the natural numbers. For other- wise, by the law of the natural numbers, 1U would include a number less than all the rest. This least number would exceed 1, for it was assumed that 1 belongs to s l)i; hence it could be denoted by n + 1. Then n would belong to 3)i, but (n + 1) would not, which contradicts the hypotheses in the law of induction. 5 In applications it is usually an advantage to be able to make state- ments not merely about the natural numbers but about any whole numbers. 6 The following rather more general form of the law of induction can be deduced in exactly the same way from the fundamental law of the natural numbers. If set >j.)j of natural numbers includes the number 1, and if the number (n -|- 1) can be proved to belong to the aggregate provided the number n does, then Wl con- tains all the natural numbers. 1. The system of rational numbers and its gaps. 7 The laws then take the following forms, obviously equivalent to those above : Law of Induction. If a statement involves a natural number n (e. g. "if n ^ 10, then 2 W > n*", or the like) and if a) this statement is correct for n = p t and b) its correctness for n = p, p -{- I, . . . , k (where k is any natural number >; p) always implies its correctness for n = k -f- 1, then the statement is correct for every natural number ^ p. Law of Integers. In every set of integers all r p that is not "empty", there is always a number less than all the rest. 6 We will lastly mention a theorem susceptible, in the domain of rational numbers, of immediate proof, although it becomes axiomatic in character very soon after this domain is left; namely the VI. Theorem of Eudoxus. If a and b are any two positive rational numbers, then a natural number n always exists 7 such that n b > a. The four ways of combining two rational numbers give in every case as the result another rational number. In this sense the system of rational numbers forms a closed aggregate (naturlicher Rationalitats- bereich or number corpus). This property of forming a closed system \\ith respect to the four rules is obviously not possessed by the aggregate of all natural numbers, or of all positive and negative integers. These are, so to speak, too sparsely sown to meet all the demands which the four rules make upon them. This closed aggregate of all rational numbers and the laws which hold in it, are then all that we regard as given, known, secured. As that type of argument which makes use of inequalities and absolute values 3. may be a little unfamiliar to some, its most important rules may be set down here, briefly and without proof: I. Inequalities. Here all follows from the laws of order and monotony. In particular 1. The statements in the laws of monotony are reversible; e. g. a -f- c < b -|- c always implies a < 6; and so does a c < b c , provided c > 0. 2. a < b, c < d always implies a -f c < b -f d. 3. a < b, c < d implies, provided b and c are positive, a c < b d. 4. a < b a! ways implies b < a, . . . 11 and also, provided a is positive, , < -. b a To reduce these forms of the laws to the previous ones, we need only con- sider the natural numbers m such that, in the one case, the statement in question is correct for n (p 1) -f m, or, in the other, that (p 1) -f m belongs to the non-" empty" set under consideration. 7 This theorem is usually, but incorrectly, ascribed to Archimedes ; it is already to be found in Euclid, Elements, Book V, Def. 4. 8 Chapter I. Principles of the theory of real numbers. Also these theorems, as well as the laws of order and monotony, hold (with appropriate modifications) when the signs "S", "-*", "__-" and <c ^= l> are sub- stituted for "<", provided we maintain the assumptions that c> b and a are posi- tive, in 1, 3, and 4 respectively. II. Absolute values. Definition: By \ a |, the absolute value (or modulus) of a, is meant that one of the two numbers -\-a and a which is positive, sup- posing a 3= 0; and the number 0, if a 0. (Hence | | -^ and if a = 0, | a \ > 0.) The following theorems hold, amongst others: 3. a\ --- \ - a\. 2. | ab \ =- 1 a a , provided a =f= 0. J 4. \ a + b [ :_j, \ a\ + \ b \; |a + 6|^|a|- |6|, and indeed | a + b \ ^ \a\ -|6|(. 5. The two relations | a \ < r and r < a < r are exactly equivalent; similarly for | x a \ < r and a r <. x < a -\- r. 0. | a b | is the distance between the points a and b, with the represen- tation of numbers on a straight line described immediately below. Proof of the first relation in 4: a ^ \ a |, b < | b |, so that by 3, I, 2, (a -\ b) ^ | a | -f- | 6 |, and hence | a -\ b \ ^ | a \ -}- | b |. We also assume it to be known how the relations of magnitude between rational numbers may be illustrated graphically by relations of positions between points on a straight line. On a straight line or number-axis, any two distinct points arc marked, one O, the origin (0) and one U 9 the unit point (1). The point P which is to represent a number a = *- (q > 0, p ^ 0, both integers) is obtained by marking off on the axis, | p | times in succession, beginning at O, the <? th part of the dis- tance O U (immediately constructed by elementary geometry) either in the direction O U, if p > 0, or if p is negative, in the opposite direction. This point 8 we call for brevity the point a, and the totality of points corresponding in this way to all rational numbers we shall refer to as the rational points of the axis. The straight line is usually thought of as drawn from left to right and U chosen to the right of O. In this case, the words positive and negative obviously become equiva- lents of the phrases: to the right of O and to the left of O, respectively; and, more generally, a < b signifies that a lies to the left of b, b to the right of a. This mode of expression may often assist us in illustrating abstract relations between numbers. 8 The position of this point is independent of the particular representation of the number a t i. e. if a p'/q' is another representation with </' *> and p' ^ both integers, and if the construction is performed with q', p' in place of q t p, the same point P is obtained. 1. The system of rational numbers and its gaps. 9 This completes the sketch of what we propose to take as the previously secured foundation of our subject. We shall now regard the description of these foundations as characterizing the concept of number; in other words, we shall call any system of conceptually well- distinguished objects (elements, symbols) a number system, and its elements numbers, if to put it quite briefly for the moment we can operate with them in essentially the same ways as we do with rational numbers. We proceed to give this somewhat inaccurate statement a precise formulation. We consider a system S of well-distinguished objects, which we denote by a, /?,.... S will be called a number system and its elements a, j3, . . . will be called numbers if, besides being capable of definition exclusively by means of rational numbers (i. c. ultimately by means of natural numbers alone) 9 , these symbols a, jS, . . . satisfy the following four conditions : 1. Between any two elements a and /3 of S one and only one of the three relations 10 a < 0, a = a > necessarily holds (this is expressed briefly by saying that S is an ordered system) and these relations of order between the elements of S are subject to the same fundamental laws 1 as their analogues in the system of rational numbers u . 2. Four distinct methods of combining any two elements of S are defined, called Addition, Subtraction, Multiplication and Division. With a single exception, to be mentioned immediately (3.), these processes can always be carried out to one definite result, and obey the same Fun- damental Laws 2, I IV, as their analogues in the system of the rational 9 We shall come across actual examples m 3 and 5; for the moment, we n.ay think of decimal fractions, or similar symbols constructed from rational numbers. See also footnote 10, p. 12. 10 Cf. also footnotes 2 and 4. 11 As to what we may call the practical meaning of these relations, nothing Is implied; "<" may as usual stand for "less than'*, but it may equally well mean "before", "to the left of", "higher than", "lower than", "subsequent to", in fact may express any relation of order (including "greater than"). This meaning merely has to be defined without ambiguity and kept consistent. Similarly, "equality" need not imply identity. Thus, for example, within the system of symbols of the form p/q, where/), q are integers and q =4= 0, the symbols 3/4, 0/8, I)/ 12 are generally said to be "equal"; that is, for certain purposes (calculating, measuring, and so on) we define equality within our system of symbols in such a way that 3/4 -= 6/8-= -9/-12, although 3/4, 0/8, -9/-12 are in the first instance different elements of that system (see also 14, note 1). 10 Chapter I. Principles of the theory of real numbers. numbers 12 . (The "zero" of the system, which must be known in order that the elements can be divided into positive and negative, is to be defined as explained in footnote 14 below.) 3. With every rational number we can associate an element of S (and all others "equal' ' to it) in such a manner that, if a and b denote rational numbers, a, ft their associates from S: a) the relation 1. holding between a and ft is of the same form as that holding between a and b. b) the element resulting from a combination of a and ft (i. e. a + ft, a ft, a ft, or a -f- ft) has for its associated rational number the result of the similar combination of a and b (i. e. a + b, a b, a b, or a -^ b respectively). [This is also expressed, more shortly, by saying that the system S contains a sub-system S' sivnilar and isomorphous to the system of rational numbers. Such a sub-system is in fact constituted by those elements of S which we have associated with rational numbers 13 .] In such a correspondence, an element of S associated with the rational number zero, and all elements equal to it, may be shortly referred to as the "zero" of the system of elements. The exception mentioned in 2. then relates to division by zero 14 . 12 With reference to these four processes it should be noted, as in the case of the symbols < and -, that no practical interpretation is implied. We also draw attention to the fact that subtraction is already completely denned in terms of addition, and division in terms of multiplication, so that, properly speaking, only two modes of combining elements need be assumed known. 13 Two ordered systems are similar if it is possible to associate each element of the one \\ith an element of the other in such a way that the same one of the relations 4, 1 as holds between two elements of the one system also holds between the two associated elements of the other, they are tsomorfihous relatively to the possible modes of combining their elements, if the element resulting from a com- bination of two elements of the one system is associated with that resulting from the similar combination of the two associated elements of the other system. 14 The third of the stipulations by means of which we here characterise the concept of number is fulfilled, moreover, as a consequence of the first arid second. For our purposes, this fact is not essential; but as it is significant from a systematic point of view, we briefly indicate its proof as follows' By 4, 2, there is an element for which a -f- a. From the fundamental laws 2, 1, it then quite eastl> follow^ tha one and the same element of S satisfies a -I- - a, for every a. This element , with all elements equal to it, is called the neutral element relatively to the process of addition, or for brevity the "zero" in S. If a is different from this "zero", there is, further, an element for which a e a; and it again appears thit this element is the same as that satisfying n - a for any other a in S. This e, with all elements equal to it, is called the neutral element relatively to the process of multiplication, or, briefly, the "unit" in S. The elements of S produced bv repeated addition or subtraction of this "unit", and any others equal to them, are then called "integers" of S. All further elements of S (and all equal to them) which result fiom these by the process of division then form the sub-system S' of S in question; that it is similar and i amorphous to the system of all rational numbers is in fact easily deduced from 4, i and 4, 2. Thus, as asserted, our concept of number is already determined by the requirements of 4, 1, 2 and 4. 1. The system ot rational numbers and its gaps. 11 4. For any two elements a and /3 of S both standing in the relation ">" to the "zero" of the system, there exists a natural number n for which n j8 > a. Here n )3 denotes the sum ]8 -f- jf? + . . . -|- ]8 containing the element ]8 w times. (Postulate of Eudoxus; cf. 2, VI.) To this abstract characterisation of the concept of number we will append the following remark l5 : If the system S contains no other elements than those corresponding to rational numbers as specified in 3, then our system does not differ in any essential feature from the system of rational numbers, but only in the (purely external) designation of the elements by symbols, or in the (purely practical) interpretation which we give to these symbols; differences almost as irrelevant, at bottom, as those which occur when we write figures at one time in Arabic characters, at another, in Roman or Chinese, or take them to denote now temperature, now velocity or electric charge. Disregarding external characteristics of notation and practical interpretation, we should thus be perfectly justified in considering the system S as identical with the system of rational numbers and in this sense we may put a = a, b --.&.... If, however, the system S contains other elements besides the above mentioned, then we shall say that S includes the system of rational numbers, and is an extension of it. Whether a system of this more com- prehensive kind exists at all, remains for the moment an open question; 15 We have defined the concept of number by a set of properties characterising it. A critical construction of the foundations of arithmetic, which is quite out of the question within the limits of this volume, would have to comprise a strict investigation as to the extent to which these properties are independent of one another, i. e. whether any one of them can or cannot be deduced from the rest as a provable fact. Further, t would have to be shuwn that none of these fundamental stipulations is in contradiction with any other and other matters too would require consideration. These investigations are tedious and have not yet reached a final conclusion. In the treatment by E. Landau mentioned on p. 2, footnote 7, it is proved with absolute rigour that the fundamental laws of arithmetic which we have set up can all be deduced from the following 5 axioms relating to the natural numbers: Axiom 1 : 1 is a natural number. Axiom 2: For every natural number n there is just one other number that is called the successor of n. (Let it be denoted by n'.) Axiom 3: We have always n' 1. Axiom 4: From m' ~~ n' t it follows that m n. Axiom 5: The induction law V is valid (in its first form). These 5 axioms, first formulated as here by G. Peano, but in substance set up by R. Dedektnd, assume that the natural numbers as a whole are regarded as given, that a relation of equality (and hence also inequality) is defined between them, and that this equality satisfies the relations 1, 1, 2, 3 (which belong to pure logic). 12 Chapter I. Principles of the theory of real numbers. but an example will come before our notice presently in the system of real numbers 16 . Having thus agreed as to the amount of preliminary assumption we require, we may now drop all argument on the subject, and again raise the question: What do we mean by saying that we know the number V2 or TT? It must in the first instance be termed altogether paradoxical that a number having its square equal to 2 does not exist in the system so far constructed 17 , or, in geometrical language, that the point A of the number-axis, whose distance from O equals the diagonal of the square of side O U, coincides with none of the "rational points". For the rational numbers are dense, i. e. between any two of them (which are distinct) we can point out as many more as we please (since, if a ^ b y fo a the n rational numbers given by a + v , for v = 1, 2, . . . , n, evi- n -|- 1 dently all lie between a and b and are distinct from these and from one another); but they are not, as we might say, dense enough to symbolise all conceivable points. Rather, as the aggregate of all integers proved too scanty to meet the requirements of the four processes of arithmetic, 16 The mode of defining the number-concept given in 4 is of course not the only possible one. Frequently the designation of number is still ascribed to objects which fail to satisfy some one or other of the requirements there laid down. Thus for instance we may relinquish the condition that the objects under con- sideration should be constructively developed from rational numbers, regarding any entities (for instance points, or distances, or such like) as numbers, provided only they satisfy the conditions 4, 1 4, or, in short, are similar and isomorphous to the system we have just set up. This conception of the notion of number, in accordance with which all isomoiphous systems must be regarded as in the ab- stract sense identical, is perfectly justified from a mathematical point of view, but objections necessarily arise in connection with the theory of knowledge. We shall encounter another modification of the number -concept when we come to deal with complex numbers. 17 Proof'. There is certainly no natural number of square equal to 2, as I 2 - 1 and all other integers have their squares ^ 4. Thus V2 could only be a (positive) fraction , where q may be taken ^ 2 and prime to p (i. e. the fraction is in its lowest terms). But if - is in its lowest terms, so is ( - J , which there- Q W/ Q ' q fore cannot reduce to the whole number 2. In a slightly different form: For any two natural numbers p and q without common factor, we have necessarily /> 2 4- 2 q~. For since two integers without common factors cannot both be even, either p is odd, or else p is even and q odd. In the first case /> 2 is again odd, hence cannot equal an even integer 2 q 2 . In the second case p 2 = (2 p'Y is divisible by 4, but 2 q z is not, since it is double an odd number. So p' 2 =1= 2 r/ 2 again. This Pythagoras is said to have already known (cf. M. Cantor, Gesch. d. Mathem., Vol. 1, 2 lj ed., pp. 142 and 169. 1894). 1. The system of rational numbers and its gaps. 13 so also the aggregate of all rational numbers contains too many gaps 18 to satisfy the more exacting demands of root extraction. One feels, nevertheless, that a perfectly definite numerical value belongs to the point A and therefore to the symbol V2. What are the tangible facts which underlie this feeling? Obviously, in the first instance, this: We do, it is true, know perfectly well that the values 1-4 or 1*41 or 1*414 etc. for V2 are in- accurate, in fact that these (rational) numbers have squares < 2, i. e. are too small. But we also know that the values 1-5 or 1-42 or 1*415 etc. are in the same sense too large; that the value which we are attempting to reach would have therefore to lie between the corres- ponding too large and too small values. We thus reach the definite conviction that the value of N/2 is within our grasp, although the given values are all incorrect. The root of this conviction can only lie in the fact that we have at our command a process, by which the above values may be continued as far as we please; we can, that is, form pairs of decimal fractions, with 1, 2, 3, ... places of decimals, one frac- tion of each pair being too large, and the other too small, and the two differing only by one unit in the last decimal place, i. e. by (y 1 ^) 71 , if n is the number of decimal places. As this difference may be made as small as ive <please, by sufficiently increasing the number n of given decimal places, we are taught through the above process to enclose the value which we are in search of between two numbers as near as we please to one another. By a metaphor, somewhat bold at the present stage, we say that through this process V2 itself is "given", in virtue of it, V2 is "known", by it, V2 may be "calculated", and so on. We have precisely the same situation with regard to any other value which cannot actually be denoted by a rational number, as for instance TT, log 2, sin 10 etc. If we say, these numbers are known, nothing more is implied than that we know some process (in most cases an extremely laborious one) by which, as detailed in the case of V2, the desired value may be imprisoned, hemmed in, within a narrower and narrower space between rational numbers, and this space ultimately narrowed down as much as we please. For the purpose of a somewhat more general and more accurate 18 This is the paradox, scarcely capable of any direct illustration, that a set of points, dense in the sense just explained, mav already be marked on the number axis, and yet not comprise all the points of the straight line. The situation may be described thus: Integers form a first rough partition into compartments; rational numbers fill these compartments as with a fine sand, which on minute inspection inevitably still discloses gaps. To fill these will be our next problem. 14 Chapter I. Principles of the theory of real numbers. statement of these matters, we insert a discussion of sequences of rational numbers, provisional in character, but nevertheless of fundamental im- portance for all that comes after. 2. Sequences of rational numbers 1 . In the process indicated above for calculating V2, successive well- defined rational numbers were constructed; their expression in decimal form was material in the description; from this form we now propose to free it, and start with the following 5. Definition. If, by means of any suitable process of construction, we can form successively a first, a second, a third, . . . (rational) number and if to every positive integer n one and only one well-defined (rational) number x n thus corresponds, then the numbers X l> X 2> X '3> > X m (in this order, corresponding to the natural order of the integers 1 , 2, 3, ... n, . . .) are said to form a sequence. We denote it for brevity by (x n ) or (*!, * 2 , . . .). O Examples. i u i * n ~~ ] '* C * sec l uence > or ] > 2' 3' 2. x n - 2"; i. e. the sequence 2, 4, 8, 16, ... 3. x n a n ; i. e. the sequence a, a 2 , a 3 , . . . , where a is a given number. - 4. x n ~ H 1 - (- 1 ) 71 }; 1- e. the sequence 1, 0, 1, 0, 1, 0, ... 6. x n = the decimal fraction for V2, terminated at the w th digit. / iyi i 111 6. x n - L_^.__ ; i. e . the sequence 1, - i, + * - ' . . . n & j * 7. Let x 1 = 1, x 2 = 1, # 3 = x l + # 2 ~ ^ and, generally, for n > 3, let x n ~ x n-i + x n-z- We thus obtain the sequence 1, 1, 2, 3, 5, 8, 13, 21, . . . , ubually called Fibonacci's sequence. 8. l,2,},-8,-J,S,J,-3,-J,... o 3 4 5 + I A 2,3,3,..., - n .... 10 1 2 3 4 - 1 10 - U '2'3'4' n""" 11. x n the w th prime number 2 ; i. e. the sequence 2, 3, 5, 7, 11, 13, ... \ 12. The sequence 1, |, ^, g, ^ m wh.ch * = (l + J + . . . + i) 1 In this section all literal symbols will continue to stand for rational numbers only. 2 Euclid proved that there is an infinity of primes. If p lt p 2 , . . . , p k are any prime numbers, then the integer m -= (/>,/> 2 . . . p k ) + 1 is either a prime different from pi, pi, . . . , p k , or else a product of such primes. Hence no finite set of prime numbers can include all primes. 2. Sequences of rational numbers. 15 Remarks. 1. The law of formation may be quite arbitrary; it need not, in particular, be embodied in any explicit formula enabling us to obtain x n , for a given n t by direct calculation. In examples 6, 5, 7 and 11, clearly no such formula can be im- mediately written down. If the terms of the sequence are individually given, neither the law of formation (cf. 6, 5 and 12) nor any other kind of regularity (cf. 6, ll) among the successive numbers is necessarily apparent. 2. It is sometimes advantageous to start the sequence with a "0 th " term x , or even with a ( l) th or ( 2) th term, x__ lt #_ 2 . Occasionally, it pays better to start indexing with 2 or 3. The only essential is that there should be an integer m ^ such that x n is defined for every n ^ m. The term x m is then called the initial term of the sequence. We will however, even then, continue to designate as the n ih term lhat which bears the index n. In 6, 2, 3 and 4, for instance, we can without further difficulties take a th term or even ( l) t}l or ( 2) <h to head the sequence. The "first term" of a sequence is then not necessarily the term with which the sequence begins. The notation will be preferably (x 0> *i> ) or (#-i #o> ) etc., as the case may be, unless it is either quite clear or irrelevant where our enumeration begins, and the abbreviated notation (x n ) can be adopted. 3. A sequence is frequently characterised as infinite. The epithet is then merely intended to emphasize the fact that every term is succeeded by other terms. It is also said that there is an infinite number of terms. More generally, there is said to be a finite number or an infinite number of things under consideration accord- ing as the number of these things can be indicated by a definite integral number or not. And we may remark here that the word infinite, when otherwise used in the sequel, will have a symbolic significance only, intended as a concise expression of some perfectly definite (and usually quite simple) circumstance. 4. If all the terms of a sequence have one and the same value c, the sequence is said to be identically equal to c, and in symbols (x n ) ~ c. More generally, we shall write (x n ) == (x n ') if the two sequences (x n ) and (x n ') agree term for term, i. e. for every index in question x n ~ x n '. 5. It is often helpful and convenient to represent a sequence graphically by marking off its terms on the number-axis, or to think of them as so marked. We thus obtain a sequence of point*. But in doing this it should be borne in mind that, in a sequence, one and the same number may occur repeatedly, even "in- finitely often" (cf. 6, 4); the corresponding point has then to be counted (i. e. con- sidered as a term of the sequence of points) repeatedly, or infinitely often, as the case may be. 0. A graphical representation of a different kind is obtained by marking, with respect to a pair of rectangular coordinate axes, the points whose coordinates are (w, x n ) for w = 1, 2, 3, ... and joining consecutive points by straight segments. The broken line so constructed gives a picture (diagram, or graph) of the sequence. To consider from the most diverse points of view the sequences hereby introduced, and the real sequences that will shortly be defined, will be the main object of the following chapters. We shall be interested more par- ticularly in properties which hold, or are stipulated to hold, for all the terms of the sequence, or at least for all terms beyond (or following) some definite term 3 . With reference to this last restriction, it may sometimes 8 E. g. all the terms of the sequence 6, 9 are > 1. Or, all the terms of the sequence 6, 2 after the 6 th are > 100 (or more shortly: for n > 6, x n > 100). 16 Chapter I. Principles of the theory of real numbers. be said that particular considerations in hand are valid "a finite number of terms being disregarded", or only concern the ultimate behaviour of the sequence. Our first examples of considerations of the kind referred to are afforded by the following definitions: Definitions. I. A sequence is said to be bounded*, if there is a positive number K such that each term x n of the sequence satisfies the inequality x n ^ K or The number K is then called a bound of the sequence. Remarks and Examples. 1. In definition 8, it is a matter of practical indifference whether we write "" or "<K". For if | x n \ ^ K holds always (i. e. for every n in question), then we can also find a constant K' such that \ x n \ < K' holds always; indeed, clearly any K.' > K will serve the purpose. Conversely, if | x n \ < K. always, then a fortiori \ x n \ ^ K. When the exact magnitude of the bound comes in of course the distinction may be essential. 2. If K is a bound of (x n ) t then so is any larger number K'. 3. The sequences 6, 1, 4, 5, 6, 9, 10 are evidently bounded; so is 6, 3, pro- vided | a | Si 1. The sequences 6, 2, 7, 8, 11 are certainly not so. Whether 6, 3 for every \a\ >1, or 6, 12, is bounded or not, i> not immediately obvious. 4. If all we know is the existence of a constant K lt such that x n < K lt for every n t then the sequence is said to be bounded on the right (or above) and K l is called a bound above (or a right hand bound) of the sequence. If there is a constant K 2 such that x n > K 2 always, then (x n ) is said to be bounded on the left (or below) and K 2 is called a bound below (or a left hand bound) of the sequence. Here K and K 2 need not be positive. 5. Supposing a given sequence is bounded on the right, it may still happen that among its numbers none is the greatest. For instance, 6, 10 is bounded on the right, yet every term of this sequence is exceeded by all that follow it, and none can be the greatest 6 . Similarly, a sequence bounded on the left need contain no least term; cf. 6, 1 and 0. (With this fact, which will appear at first sight para- doxical, the beginner should make himself thoroughly familiar.) Among a finite number of values there is of course always both a greatest and a least, i. e. a value not exceeded by any of the others, and one which none of the others falls below. (There may, however, be several equal to this greatest or least value.) (5. The property of boundedness of a sequence x n (though not the actual value of one of the bounds) is a property of the tail-end of the sequence ; it is unaffected by any alteration to an isolated term of the sequence. (Proof?) 4 This nomenclature appears to have been introduced by C. Jordan, Cours d'analyse, Vol. 1, p. 22. Paris 1893. 6 The beginner should guard against modes of expression such as these, which may often be heard: "for n infinitely large, x n 1"; "1 is the greatest number of the sequence". Anything of this sort is sheer nonsense (cf. on this point 7, 3). For the terms of the sequence are 0, ,],},... and none of these is -- 1, on the contrary all of them are < 1. And there is no such thing as an "infinitely large n". 2. Sequences of rational numbers. 17 II. A sequence is said to be monotone ascending or increasing 9. if, for every value of n, X n ^ X n+ il it is said to be monotone descending or decreasing if, for every n, x n S X n +l* Both kinds will also be referred to as monotone sequences. Remarks and Examples. 1. A sequence need not of course be either monotone increasing, or mono- tone decreasing; cf. 6, 4, 6, 8. Monotone sequences are, however, extremely com- mon, and usually easier to deal with than those which are not monotone. That is why it is convenient to give them a distinguishing name. 2. Instead of "ascending" we should more strictly say "non-descending", and instead of "descending", "non-ascending". This, however, is not customary. If in any special instance the sign of equality is excluded, so that x n < x ni l or v n > x n} ,, as the case may be, for every n t then the sequence is said to be strictly monotone (increasing or decreasing). 3. The sequences 6, 2, 5, 7, 10, 11, 12 and 6, 1, 9 are monotone; the first- named ascending, the others descending. 6, 3 is monotone descending, if ^ a ^ 1, but monotone ascending if a " . 1 ; for a < 0, it is not monotone. 4. The designation of "monotone" is due to C. Neumann (Ober die nach Kteis-, Kugel- und Zylmderfunktionen fortschreitenden Entwickelungen, pp. 2(5, 27. Leipzig 1881). We now come to a definition to which the reader should pay the greatest attention, sparing no effort to make himself master of its meaning and all that it implies. III. A sequence will be called a null sequence if it possesses the fol- 10 lowing property: given any arbitrary positive (rational) number e, the in- equality | x n | < c is satisfied by all the terms, with at most a finite number 6 of exceptions. In other words : an arbitrary positive number e being chosen, it is always possible to designate a term x m of the sequence, beyond which the terms are less than e in absolute value. Or a number n Q can always be found, such that |*|< for Remarks and Examples. 1. If, in a given sequence, these conditions are fulfilled for a particular e, they will certainly be fulfilled for every greater e (cf. 8, 1), but not necessarily for any smaller e. (In 6, 10, for instance, the conditions are fulfilled for e = 1 and there- fore for every larger e, if we put n =0; for e - } it is not possible to satisfy them.) In the case of a null sequence, the conditions have to be fulfilled for every positive 8 Cf. 7, 3. 18 Chapter I. Principles of the theory of real numbers. , and in particular, therefore, for every very small e > 0. On this account, it is usual to formulate the definition somewhat more emphatically as follows: (x n ) is a null sequence if, to every > 0, however small, there corresponds a number n such that | x n | < c for every n > n . I x n | < c, provided n > whatever be the value of e. It is thus sufficient to put n a Here w () need not be an integer. 2. The sequence 6, 1 is clearly a null sequence; for -, . 3. The place in a given sequence beyond which the terms remain numeri- cally < e, will naturally depend in general on the magnitude of e; speaking broadly, it will lie further and further to the right (i. e. n n will be larger and larger), the smaller the given c is (cf. 2). This dependence of the number n on e is often emphasised by saying explicitly: "To each given corresponds a number n Q w (t) such that ..." 4. The positive number below which | x n \ is to he from some stage onwards need not always be denoted by c. Any positive number, however designated, may serve. In the sequel, where e, a, K t . . . , denoting any given positive numbers, we may often use instead ^, ^, ^, e 2 , a e, t a , etc. 5. The sign of x n plays no part here, since | x n \ = | x n \. Accordingly 6, is also a null sequence. 6. In a null sequence, no term need be equal to zero. But all terms, whose index is very large, must be very small. For if I choose e = 10~~, say, then for cver\ n > a certain n 0t ( x n \ must be < 10~' 5 . Similarly for e - - 10~ 10 and for any other e. 7. The sequence (a n ) specified in 6, 3 is also a null sequence provided \ a \ < 1. Proof. If a 0, the assertion is trivial, since then, for every > 0, | x n \ < for every n. If < | a \ < 1, then (by 3, 1,4). ---. > 1. If therefore we put I * I * = 1 4- p t then p > 0. I a \ But in that case, for every n ^ 2, we have (a) <l + #) n >! + # For when n = 2, we have (1 4- />) 2 ^ 1 + 2/> -f p z > 1 -f 2p; the stated relation therefore holds in that case. If, for n k ^ 2, (!+/>)*> 1-1- kp, then by 2, III, 6 therefore our relation, assumed true for n = k t is true for w = & + 1. By 2, V it therefore holds 7 for every n ^ 2. 7 The proof shows moreover that (a) is valid for n ^ 2 provided only 1 4- P > 0, i. e. p > 1, but =t=0. For p -- and for n = 1, (a) becomes an equality. For /> > 0, the validity of (a) follows immediately from the expansion of the left- hand side by the binomial theorem. The relation (a) is called Bernoulli's Inequality (James Bernoulli, Propositiones arithmeticae de seriebus, 1689, Prop. 4). 2. Sequences of rational numbers. 19 Accordingly, we now have so that, however small c > may be, we have I x n I ^ I aU I < for every n > P 8. In particular, besides the sequence f ) mentioned in 2., ( -), (-- J, //4\"\ . W \ 2n / \ 3fi / ( (?) )' i ui 9. A similar remark to that of 8, 1 may be appended to Definition 10: no essential modification is produced by reading "5* e" for "< e" there. In fact, if, for every n > w () , | x n \ < e, then a fortiori \ x n \ 5^ c; conversely, if, given any e, ?2 can be so determined that | x n \ '^ e for every n > w 3 , then choosing any posi- tive number e t < c there is certainly an n 1 such that | x n \ fg c l9 for every n > n^ and consequently | x n | < for every n > n t ; the conditions in their original form are thus also fulfilled. Precisely analogous considerations show that in Definition 10 "> H O " and "^ w " are practically inter- changeable alternatives. In any individual case, however, the distinction must of course be taken into account. 10. Although in a sequence every term stands entirely by itself, with a definite fixed value, and is not necessarily in any particular relation with the preceding or following terms, yet it is quite customary to ascribe "to the terms x n ", or "to the general term' 1 any peculiarities in the sequence which may be observed on running through it. We might say, for instance, in 6, 1 the terms diminish; in 6, 2 the terms increase; in 6, 4 or 6, 6 the terms oscillate; in 6, 11 the general term cannot be expressed by a formula, and so on. In this sense, the character- istic behaviour of a null sequence may be described by saying that the terms become arbitrarily small, or infinitely small 8 ; by which neither more nor less is meant than is contained in Definition 9 10, viz. that for every > however small the terms are ultimately (i. e. for all indices n > a suitable n ; or from and after, or beyond, a certain n (t ) numericallv less than e. 11. A null sequence is ipso facto bounded. For if we choose e I, then there must be an integer n, such that, for every n > n i9 \ x n \ < 1. Among the finite number of values | .v t |, | x 2 1, . . . , ! .v nl |, however, one (cf. 8, 5) is greatest, M say. Then for K M -f 1, obviously | .v w | is akvays < K. 12. To prove that a given sequence is a null sequence, it is indispensable to show that for a prescribed e > 0, the corresponding w y can actually be proved to exist (for instance, as in the examples that follow, by actually designating such a number). Conversely, if a sequence (x n ) is assumed to be a null sequence, it is thereby assumed that, for every t, the corresponding n may really be regarded as existent. On the other hand, the student should make sure that he understands clearly what is meant by a sequence not being a null sequence. The meaning is this : it is not true that, for every positive number *, beyond a certain point | x n \ 6 This mode of expression is due to A. L. Caitchy (Analyse algebrique, pp. 4 and 2G). 9 There need of course be no question here of the sequence being monotone. Also, in any case, some | x n | 's of index 5* w may already be < c. 20 Chapter I. Principles of the theory of real numbers. is always < e; there exists a special positive number e,,, such that | x n \ is not, beyond tiny // , always < c () ; after every // there is a larger index n (and therefore an in- finite number of such indices) for which | v n | ]> c . 1 3. Finally we may indicate a means of interpreting geometrically the special character of a null sequence. Using the graphical representation 7, 5, the sequence is a nuii sequence if its terms ultimately (for n > n n ) all belong to the interval 10 e . . . -f- . Let us call such an interval for brevity an e-neighbourhood of the origin; then we may state (x n ) is a null sequence if every c-neighbourhood of the origin (however small) contains all but a finite number, at most, of the terms of the sequence. Similarly, using the graphical representation 7, 6, we can state: (x n ) is a null sequence if every *-stnp (however narrow) about the a\ts of absci^ae contains the entire graph, with the exception, at most, of a finite initial portion, the e-strip being limited by parallels to the axis of abscissae through the two points (0, e). 14. The concept of a null sequence, the "arbitrarily small given positive number c", to which we shall from now on have continually and indispensably to appeal, and which may thus be said to form a main support for the whole super- structure of analysis, appears to have been first used in 1055 by J. Walks (v. Opera I., p. 3S2/3). Substantially, however, it is already to be found in Euclid, Elements V. We are already in a better position to comprehend what is involved in the idea, discussed above, of a meaning for V2 or TT or log 5. In forming on the one hand (we keep to the instance of V2) the numbers * 1 =l-4; *o=l-41; * a = 1-414; * 4 == 1-4142; ... on the other, the numbers yi = I'O; y* - 1-42; ^ - 1415; y, =-- 1-4143; . . . we are obviously constructing two sequences of (rational) numbers (x n ) and (y n ) according to a perfectly definite (though possibly very laborious) method of procedure. These two sequences are both monotone, (x n ) increasing, (y n ) decreasing. Furthermore x n is <y n for every //, but the differences, i. e. the numbers y n x n =- d n form, by 10, 8, a null sequence, since d n = n . These are clearly the facts which convince us that we "know" V2, and can "calculate" it ? and so on, although as we said before no one has yet had the value V2 completely within his view, so to speak. If we refer again to the more suggestive representation on the number-axis, then, obviously (cf. fig. 1, p. 25): the points x l and y determine an interval 10 The word interval denotes a portion of the number-axis between a definite pair of its points. According as we reckon these points themselves as belonging to the interval or not, this is termed closed or open. Unless otherwise stated, the interval will always in the sequel be regarded as closed. (For 10, 13 this is immaterial, by 10, 9.) Supposing a to be the left end point, b the right end point, of an interval, we call this for brevity the interval a ... b. 2. Sequences of rational numbers. 21 ! of length d l ; the points x 2 and jy 2 similarly, an interval / 2 of length . Since the second interval lies wholly within the first. Similarly, the points X 3 and V 3 determine an interval of length d 3 , completely within / 2 , and generally, the points x n and y n determine an interval f n completely inside J n - V The lengths of these intervals form a null sequence; the intervals themselves shrink up, one surmises, about a definite number, contract to a quite definite point. It only remains to examine how near this surmise is to truth. With this purpose in view, we state, more generally, the following: Definition. To express the fact that a monotone ascending sequence 11. (x n ) and a monotone descending sequence (y n ) are given, whose terms for every n satisfy the condition x n ^y n and for which the differences dn=y n - Xn form a null sequence, we say for brevity that we are given a nest of intervals (Intervallschachtelung)*. TJie n th interval stretches from x n to y n and has length d n . The nest itself will be denoted by ( /) or by (# | y n )- The conjecture which we made above now finds its first confirma- tion in the following: Theorem f . There is at most one (rational) point s belonging to all 12. the intervals of a given nest, that is to say satisfying, for every n t the in- equality *n^s^ y n > Proof: If there were, besides $, another number s f differing from it, and also satisfying the inequality for every , then, for every , besides x n <Ls< y n , * A set or series of similar objects is said to form a nest or to be nested (inein- ander geschachtelt) when each smaller one is enclosed or fits into that which is next in size to it. The word nest is here used with the additional (ideal) characteristic implied, that the sizes diminish to zero. When this is not implied, we shall use the more explicit phrase that each is contained in the preceding (or we might say that they are nested). f We note here for future reference that this theorem continues to hold un- altered when the numbers which occur are arbitrary real numbers. 2 (051) 22 Chapter I. Principles of the theory of real numbers. we should also have (v. 3, I, 4) by 3, I, 2 and 3, II, 5, the inequalities would therefore hold for every n. Choosing = | s s r |, d n would never (a fortiori not for every n beyond a certain // ) be < . This contradicts the hypothesis that (d n ) is a null sequence. The assumption that two distinct points belong to all the intervals is therefore inadmissible 11 . Q. E. D. Remarks and Examples. 1. Let* n = "-"--, y = ^J; that is to say,/,, - 5J=J . . . "-J- 1 , d n = ? We can at once verify that we actually have a nest of intervals here, since 2 x n ^ x n+i "^ yn+i ^ Vn ^ or ever y n t an< ^ since, for every n > , we have d n < t however > be chosen. The number 5=1 here belongs to all the / 's, since n ~~- - < 1 < - ~ * n n n for every n. No number other than 1 can belong therefore to all the intervals. 2. Let f n be defined as follows 12 : / is the interval ... 1; / l the left half of A; Jz the right half ofy^ y 3 the left half ofy 2 ; and so on. These intervals are obviously each contained in the preceding; and since J n has length d n k>n , tmd these numbers form a null sequence, we have a nest of intervals. A little considera- tion shows that the sequence of the x n 's consists of the numbers 0> 4' 4 10 ~~ 16' 4 + T6 ~*~ G4 "" 6T * * ' each taken twice running; and that the sequence of y n 's begins with 1 and con- tinues with 1 ~" 2 = 2 J l ~ 2 ~ 8 ^ 8' ~ 2 "" 8 ~~ 32 ^ 32* * ' * each taken twice running. Now 1,1,1, , 1 1 A 1\ ^ 1 4 16 Ci ' ' ' P = 3 ~ 4*- 3 ~ 4*-) 11 From a graphical point of view, what the proof indicates is that if $ and $' belong to all the intervals, then each interval has a length at least equal to the distance | s s' | between s and s' (v. 3, II, 6); these lengths cannot, therefore, form a null sequence. 12 Here we let the index start from (cf. 7, 2). 13 For any two numbers a and b, and every positive integer k t the formula a fc - b k = (a - b)(a k ~ l + a k ~* b+ ... + a b k ~ 2 + * fc ~ 1 ) is known to hold. Whence, more particularly, for a =t= 1, the formulae 1 + a + . . . + a k ~* = ! ~ ** and a + a* + . . . + a k = \ " a * . a. l o 1 a 3. Irrational numbers. 23 Hence, for every n t x n < J < y n \ thus s J is the single number which belongs to all the intervals. Here, therefore, (/ n ) "defines" or " determines* ' the number i, or (y n ) shrinks up to the number J. 3. vf we are given a nest of intervals (/ n ), and a number s has been recog- nised as belonging to all the / n 's, then by our theorem, 5 is quite uniquely deter- mined by ( / n ). We therefore say, more pointedly, that the nest (/ n ) "defines" or "encloses" the number s. We also say that 5 is the innermost point of all the intervals. 4. If s is any given rational number and we put, for n 1 , 2, . . . , x n ~ s 1 n and y n s + -, then (x n \ y n ) is evidently a nest of intervals determining the number s itself. But this is also the case if we put, for every n, x n -^ s and y n s. Mani- festly, we can, in the most various ways, form nests of intervals defining a given number. This theorem, however, only confirms what we may regard as one half of our previously described impression; namely, that if a number s belongs to all the intervals of a nest, then there is none other besides with this property, s is uniquely determined by the nest. The other half of our impression, namely, that there must also always be a (rational) number belonging to all the intervals of a nest, is erroneous^ and it is precisely this fact which will become our induce- ment for extending the system of rational numbers. This the following example shows. As on p. 20, let x l 14; x.> 1-41 ; . . .; y l 1 >; y z = 1-42; . . . Then there is no rational number s> for which x n !L A "? y n for every n. In fact, if we put v ' v a v 7 v 2 x n x n Vn ~ 3>n then the intervals / n ' x n ' . . . y n ' also form a nest 11 . But x n f x^ < 2 for all n, and y n ' -- y n 2 > 2 for all n (because this was how x n and y n were chosen), i. e. x n f < 2 < y n '. On the other hand, if x n ;< s ^-_ y n we should have, by squaring (as we may, by 3, 1, 3), x n ' ? s 2 ^ y n ' for all n. By our theorem 12 this would in- volve s 2 = 2, which is however impossible, by the proof given in footnote 17 on p. 12. Here, therefore, there is certainly no (rational) number belonging to all the intervals. In the following paragraphs, we will investigate what, in a case such as this, should be done. 3. Irrational numbers. We must come to terms with the fact that there is no rational number whose square is 2, that the system of rational numbers is too defective, too incomplete, too full of gaps, to furnish a solution for the 14 For it follows from x n ^ x n l < y n+ i ^ y n since all the numbers are positive, so that squaring (cf. 3, I, 3) is allowed that x n ' ^ *v' n+1 < y' nf i ^ y n '; further y n ' x n ' -- (y n + x n )(y n .v n ); therefore, since .v r} and y n are certainly < 2 for every n, y n ' x n ' < --^ n , i. e. < s, provided J )n < ; and this, by 10, 8, is certainly the case for every n > a certain w . 24: Chapter I. Principles of the theory of real numbers. equation x 2 2. Indeed, this is only one of many equations for whose solution the material of the system of rational numbers proves insufficient. Almost all the numerical values which we are in the habit of denoting by \/n t log n, sin a, tan a and so on, are non-existent in the system of rational numbers and can no more be immediately "obtained", or "deter- mined", or be "stated in figures", than can V2. The material is too coarse for such finer purposes. The considerations brought forward in the preceding paragraphs point to means for providing ourselves with more suitable material. We saw, on the one hand, that, behind the conviction that we do know V2, there lay no more, substantially, than the fact that we possess a method by which a perfectly definite nest of intervals may be obtained ; for its construction, the solution of the equation x 2 2 of course gave the occasion lr> . We saw, on the other hand, that if a nest (/ n ) encloses any number s capable of specification at all (this still implying that it is a rational number) then this number s is quite uniquely defined by the nest ( / n ), - so unambiguously, indeed, that it ia entirely indifferent, whether I give (write down, indicate) the number directly, or give, instead, the nest (/) with the tacit addition that, by the latter, I mean precisely the number s which it uniquely encloses or defines. In this sense, the two data (the two symbols) are equivalent, and may to a certain extent be considered equal 16 , so that we may write in- deed: (/n) = * or (x n | y n ) = s. 15 The kernel of this procedure is in fact as follows: We ascertain that I 2 < 2, 2 2 > 2, and accordingly put # 1, y ~ 2. We then divide the interval k J Q =- x . . . y into 10 equal parts, and taking the points of division, 1 + , for k -= 0, 1, 2, . . . , 9, 10, determine by trial whether their squares are > 2 or < 2. We find that the squares corresponding to k 0, 1, 2, 3, 4 are too small, those corresponding to k = 5 y G, . . . , 10 too large, and accordingly we put Xi =1-4 and y t == 1-5. Next, we divide the interval /j. x l . . . y l into 10 equal parts, and go through a similar test with regard to the new points of division and so on. The known process for extracting the square root of 2 is intended mainly to make the successive trials as mechanical as possible. The corresponding treatment of, for instance, the equation 10* = 2 (i. e. determination of the common logarithm of 2) involves the following nest of intervals: Since 10 < 2, 10 l > 2, we here pu: X Q = 0, y = 1 and divide / = # . . . y into 10 equal parts. For the points of division, lftt we next test whether 10*/ 10 < 2 or > 2, that is to say, whether 10 fc < 2 10 or > 2 10 . As a result of this trial, we shall have to put x^ ~ 0-3, y^ ^ 0-4. The interval / l x l . . . y l is again divided into 10 equal parts, the same pro- 3 k cedure instituted for the points of division ^ -}- . and, in consequence, x z put equal to 30 and y a to 31 and so on. This obvious procedure is of course much too laborious for practical calculations. 16 The justification for this is provided by Theorems 14 to 19. 3. Irrational numbers. 25 Consequently, we will not say merely: "the nest (/ n ) defines the number s" but rather "(/) is only another symbol for the number $", or in fine, "(/ n ) is the number s" exactly as we are used to look upon the decimal fraction 0-333 ... as merely another symbol for the number , or as being precisely the number itself. It now becomes extremely natural to introduce tentatively an analogous mode of expression with regard to those nests of intervals which contain no rational number. Thus if x n , y n denote the numbers constructed previously in connection with the equation x 2 = 2, one might seeing that in the system of rational numbers there is not a single one whose square =2 decide to say that this nest (x n \ y n ) determines the "true" "value of V2 " though one incapable of being symbolised by means of rational numbers, that it encloses this X U -J J Fig. 1. value unambiguously in fine, "it is a newly created symbol for this number", or, for brevity, "it is the number itself". And similarly in every other case. If (/ n ) (x n \ y n ) is any nest of intervals and no rational number s belongs to all its intervals, we might finally resolve to say that this nest encloses a perfectly definite value, though one incapable of being directly symbolised by means of rational numbers, it deter- mines a perfectly definite number, though one unfortunately non- existent in the system of rational numbers, it is a newly created symbol for this number, or briefly: is the number itself; and this number, in contradistinction to the rational numbers, would then have to be called an irrational number. Here certainly the question arises: Can this be done without further justification ? Is it allowable ? May we, without more ado, designate these new symbols, the nests (x n \ y n ), as numbers? The fol- lowing considerations are intended to show that to this course there is no obstacle whatever. In the first instance, a simple graphical illustration of these facts on the number-axis (see fig. 1) gives every appearance of justification to our resolution. If, by any construction, we have marked a point P on the number-axis (e. g. by marking off to the right of O the length 26 Chapter I. Principles of the theory of real numbers. of the diagonal of a square of side O U) then we can in any number of ways define a nest of intervals enclosing the point P. We may do so in this way, for instance. First of all we imagine all integers ^ marked on the axis. Of these, there will be exactly one, say p, such that our point P lies in the stretch from p inclusive to (/>+!) exclusive. Accordingly we put x -= p, y p + 1, and divide the interval J Q = x . . . y Q into 10 equal parts 17 . The points of division k are p + - (with k = 0, 1, 2, . . . , 10), and among them, there will again k k be exactly one, say p + - J , such that P lies between x t p -[- * inclusive and y^ = p + * -y~ exclusive. The interval J^ x l . . . y^ is again divided into 10 equal parts, and so on. If we imagine this process continued indefinitely, we obtain a perfectly definite nest (J n ) all of whose intervals J n contain the point P. No other point P' besides P can lie in all the intervals J n . For, if that were so, all the intervals would have to con- tain the whole stretch PP', which is impossible, as the lengths of the intervals (j n has length J form a null sequence. For every arbitrarily given point P on the number-axis (rational or not) there are thus nests of intervals obviously, indeed, any number of such nests which contain that point and no other. And in the present instance, i. e. in the graphical representation on the number- axis the converse appears most plausible; if we consider any nest of intervals, there seems to be always one point (and by the reasoning above, only this one) belonging to all its intervals, which is thus deter- mined by it. We believe, at any rate, that we may infer this directly from our conception of the continuity, or gaplessness y of the straight line 18 . Thus in this geometrical representation we should have complete reciprocity: every point can be enclosed in a suitable nest of intervals and every such nest invariably encloses one and only one point. This gives us a high degree of confidence in the adequacy of our resolve to consider nests of intervals as numbers, which we now for- mulate more precisely as follows: 13. Definition. We will say of every nest of intervals (J n ) or (x n \ y n ), that it defines or, for brevity, it is, a determinate number. To represent 17 Instead of 10 we may of course take any other integer ^ 2. For furthei detail, see 5. 18 The proposition, by which the "continuity of the straight line" is expressly postulated for a proof cannot be here expected, since it is essentially a description of the form of our concept of the straight line which is involved is called the Cantor-Dedekind axiom. 3. Irrational numbers. 27 it y we use the symbol denoting the nest of intervals itself, and only as an ab- breviation replace this by a small Greek letter, writing in this sense 19 , e. g. (J n ) or (x n \y n ) - a. Now, in spite of all we have said, this cannot but seem a very arbi- trary step, the question has to be repeated most insistently: will it pass without further justification? These purely ideal objects which we have just defined these nests of intervals (or else that still extremely questionable 'something' which such a nest encloses or determines) can we speak of these as numbers? Are they after all numbers in the same sense as the rational numbers, more precisely, in the sense in which the number concept was defined by our conditions 4? The answer can only consist in deciding, whether the totality or aggregate of all conceivable nests of intervals, or of the symbols (/ n ) or ( x n \ yn) r <* introduced to denote them, forms a system of objects satis- fying these conditions 4 20 ; a system therefore to recapitulate these conditions briefly whose elements are derived from the rational numbers, and 1. are capable of being ordered; 2. are capable of being combined by the four processes (rules), obeying at the same time the fundamental laws 1 and 2, I IV; 3. contain a sub-system similar and isomorphous to the system of rational numbers; and 4. satisfy the Postulate of Eud- oxus. If and only if the decision turns out to be favourable, all will be well; our new symbols will then have vindicated their numerical char- acter, and we shall have established that they are numbers, whose totality we shall then designate as the system or set of real numbers. Now the decision in question does not present the slightest diffi- culty, and we may accordingly be brief in expounding the details: Nests of intervals or our new symbols (x n \ y n ) are certainly constructed by means of rational number-symbols alone; we have there- fore only to settle the points 4, 1 4. For this, we shall go to work in the following way: Certain of the nests of intervals define a rational number 21 , something, therefore, for which both meaning and mode of combination have been previously established. We consider two such rational- valued nests, say (x n \ y n ) s and (x n f \ y n ') = s'. With the two rational number-symbols s and s', we can immediately distinguish whether the first s is <, = or > the second s'; and we can combine the two by the four processes of arithmetic. Essentially, what we have to do is to endeavour directly to recognise the former fact, and to carry out the latter processes, on the two nests of intervals themselves by which s and s' were 19 <7 is an abbreviated notation for the nest of intervals ( / n ) or (x n \ y n ). 20 The reader should here read these conditions through again. 81 We will describe such nests for brevity as rational-valued. 28 Chapter 1. Principles of the theory of real numbers. given, and finally to extend the result to the aggregate of all nests of intervals. Each provable proposition (A) relating to rational-valued nests will ac- cordingly give rise to a corresponding definition (B). We begin by setting down concisely side by side these pairs of propositions (A) and definitions (B) 22 . 14. Equality: A. Theorem. If(x n \y n ) = 5 and (x n f \y n ') = s' are two rational-valued nests of intervals, then s = s' holds if, and only if, besides *n ^ y n and x n ' <^ y n ' 9 we have 23 for every n. On this theorem we now base the following: B. Definition. Two arbitrary nests of intervals cr (# n |j> n ) and a .= (x n f | y n ') are said to be equal if and only if or every n. Remarks and Examples. 1. The numbers x n and \ n ' on the one hand, y n and y n ' on the other, need of course have nothing whatever to do with one another. This is no more sur- prising than that rational numbers so entirely different in appearance as , g'A, and 375 should be referred to as "equal". Equality is indeed something which 22 The import of proposition and definition should in each case be interpreted in relation to the number-axis. 23 Into the very simple proofs of the propositions 14 to 19 we do not propose to enter, for the general reasons explained on p. 2. They will not present the slightest difficulty to the reader, once he has mastered the contents of Chapter II, whereas at this stage they would appear to him strange; moreover they will serve as exercises in that chapter. Merely as a specimen and example for the solution of those problems, we will here prove Theorem 14: a) If s = s' t then we have both x n ^ $ ^ y n and x n ' ^ s ^ y n ' y whence at once, x n < y n ' and x n ' ^ y^ for every n. b) If conversely x n 5$ y n ' for every n, then s ^ s' must hold. For if we had s > s', i. e. s s' > 0, then, since (y n x n ) is a null sequence, we could so choose the index p, that y p - x p < s s/ r X P - s ' > y* - * As however s is certainly ^ y p , this would imply x p s' > 0. We could therefore choose a further index r for which y/ - */ < * - s'. Since x r ' ^ $', this would imply y r ' < x^ Choosing an integer m exceed- ing both p and r, we could deduce, in view of the respective ascending and descend- ing monotony of our sequences of numbers, that a fortiori y m ' < x m , which con- tradicts the hypothesis that x n ^ y^ for every n. Thus s ^ $' is ensured. By interchanging throughout the above proof the accented and non-accented letters, we deduce in the same manner that if x n ' < y n for every n, then s' ^ s If then we have both x n ' ^ y n and x n y n ' holding for every , then s ~ s necessarily follows. Q. E. D. 3. Irrational numbers. 29 is not fixed a priori, but needs to be established by some form of definition, and it i> perfectly compatible \vith marked dissimilarity in a purely external aspect. 2. The two nests I ^ 3 ) anc * *^ ~ are ct l ua l m accordance with our present definition 3. By 14, we may write e. g. (s s -\- J = s --= (s \ s), the latter symbol denoting a nest all of whose intervals ha\e both their left and their right endpomts s. In particular, f - (0 | 0) = 0. w/ 4. It still remains to establish but the proof is so simple that vve will not go into it further that (cf. Footnote 23), in consequence of our definition, we have a) a a (Footnote 24), b) a -= a' always implies a' = a, and c) a a 7 , a' a" involve a = a". Inequality: A. Theorem. If (x n \ y n ) = s and (x n f \ y n ') s' are 15 two rational-valued nests, then we have s < s', if and only if x n ^ y n ' for every //, but not x n f 5^ y n for every ;/, * e - y>n < x m f or <** feast one M. B. Definition. Given any two nests of intervals a = (x n \ y n ) and a (x n r | y n '), then we shall say a < o-', if x n f y n ' for every ;/, but not x n ' ^ y n f or every n, i. e. for at least one m, y m -- x m '. Remarks and Examples. 1. It is clear that by 14 and 15 the totality of all conceivable nests is ordered. For if a and a' are any two of them, either there is equality, a a 7 , or, for at least one p, we have y v <* .Vj/, implying a < a 7 , or finally, for at least one r, y r ' < .v |f implying a' < a. The last two cases cannot occur simultaneously, since, for m greater than r and />, we should then have, a fortiori, v ?/ / <. v 7/1 ', which is impossible. Thus between a ard a' one and only one of the three relations always holds, and the totality of these new symbols is thus ordered by 14 and 15. 2. Here again it would have to be established in all detail that the laws of order 1 continue to hold good with the adopted definitions of equality and in- equality. Taking as model the proof in the footnote to Theorem 14, this presents so few essential difficulties that we will not enter into it further: The laic* of order do, effectually^ all remain valid. 3. In consequence of 14 and 15 we now have, therefore, for every n A n < c y n . What does this mean r It means that each of the rational numbers x n is, in ac- cordance with 14 and 15, not greater than the nest a ~ (x n \ y n ). Or: if we con- 24 Here it may be clearly recognised that this "law" is by no means trivial: it has indeed to be proved that with the given definition of equality every nest of intervals is effectually "equal" to itself, that is to say that the conditions of that definition are fulfilled, when the same nest is taken for both of the nests of intervals which we are comparing. 30 Chapter I. Principles of the theory of real numbers. sider any particular one of the numbers x n> say x p , and denote it for brevity by x, then we may write (see 14, Rem. 3) (v ;) -) x - x - - x + f j or - (x | x) and our statement takes the form (*!*) <.!*,). We may prove it as follows. If it were not true, then for at least one r, y r < x, i. e. y r < x^ and so a fortiori, if m is greater than r and p y y m < *m. which certainly cannot be the case. In the same way we see that a < y n . Accord- ingly, a is to be regarded as lyin^ between x n and y n for each n, in other word*, v con- tained within the interval J n . The fact that no other number a', besides a, can possess the same property is now easily proved. If in fact there were a second nest of intervals a' - (\ n ' \ y n ') such that for every definite index /> we also had x p ^ a' < y p , then the left hand inequality means, more precisely (cf 3), that (v^ | v p ) r^ (v n ' | y n ') and so, by 14 and 15, x p ^ y n ' for every n. Since this must hold in particular for // p, we deduce x 9 ^1 y v ' for every p, which signifies, by 14 and 15, that a ^ a'. In the same manner the right hand inequality is seen to imply that a' jj <* Thus neces- sarily a a', which was what we set out to prove. 4. By 15, a is > 0, i. e. "positive", if and only if (x n \ y n ) > (0 | 0), that is to say, if for some suitable index p, x v > 0. But in this case, as the .v w f s increase with n, we have a fortiori x n ^ for every n > p. We may therefore* say : a (v n | y n ) is positive if, and only if, all the endpomts ,v w , y n are positive from and after a definite index. The exact analogue holds of course for a < 0. 5. If or > 0, and, for every n ^ p, x n > 0, let us form a new nest (x n ' \ y n ') = a' by putting x x\ . . . *V-i all equal to x p , but every other x n ' and y n ' equal to the corresponding x n and y n . By 14, obviously a a'; and we may say: If a is positive, then there are always nests of intervals equal to it, for which all the endpoints of intervals are positive. The exact analogue holds for a < 0. So far then, in respect of the possibility of ordering them, our nests of intervals may be said to vindicate their character as numbers com- pletely. It is no more difficult to establish a similar conclusion with regard to the possibilities of combining them. 16. Addition: A. Theorem 2r> . If (x n \y n ) and (x n '\y n f ) are any two nests of intervals, then (x n + # n '> yn + y n ') w also one, and if the former are both rational-valued and respectively = s and = s\ then the latter is also rational- valued, and determines the number s + s' '. B. Definition. If (x n \ y n ) a and (x n f \ y n ') ~ &' are any two nests of intervals and a" denotes the nest (x n + x n ', y n + y n ') deduced from them, then we write a" = a + a' and a" ts called the sum of a and a'. 18 With regard to the proof, cf. footnote 23. 3. Irrational numbers. 31 Subtraction: A. Theorem. If (x n \ y n ) is a nest of intervals, then so 17. is ( y n | x n ); and if the former is rational-valued s, then the latter is also rational-valued, and determines the number s. B. Definition. If a = (x n \ y n ) is any nest of intervals and a' de- note the nest of intervals ( y n \ x n ) t we write a' = -a and say v is the opposite of cr. By the difference of two nests of inter- vals we then mean the sum of the first and of the opposite of the second. Multiplication: A. Theorem. If(x n \ y n ) and (x^ \ y n ') are any two 18. positive nests of intervals, replaced, if necessary, (in accordance with 15, 5) by two nests of intervals equal to them, for which all the endpoints of intervals are positive (or at least non-negative), then (x n x n r \y n y n ') is also a nest of intervals; and if the former are rational-valued and respec- tively s and = s', then the latter is also rational-valued, and determines the number s s'. B. Definition. If (x n \ y n ) a and (x n r \ y n f ) a are any two positive nests of intervals for which all the endpoints of intervals are positive which is no restriction, by 15, 5 and a" denote the nest (x n x n ' \y n y n ') derived from them, then we write a" = <T- a' and call o-" the ^product of a and cr'. The slight modifications which have to be made in this definition if one or both of a and or' are negative or zero, we leave to the reader, and henceforth consider the product of any two nests of intervals as defined. Division: A. Theorem. // (x n \ y n ) is any positive nest of intervals 19. for which all endpoints of intervals are positive, (cf. 15, 5) then so is ( J; Vn x n' and if the former is rational-valued, and = s, the latter is also rational- valued, and determines the number -. B. Definition. If (x n \ y n ) = a is any positive nest of intervals for which all endpoints are positive, and a' denote the nest (-- ), then we \y n xj write and say a' is the reciprocal of a. By the quotient of a first by a second positive nest of intervals we then mean the product of the first by the reciprocal of the second. The slight modifications necessary in this definition, if a (in the one case) or the second of the two nests of intervals (in the other) is negative, 32 Chapter I. Principles of the theory of real numbers. we may again leave to the reader, and henceforth consider the quotient of any two nests of intervals of which the second is different from 0, as defined. If (x n \ y n ) a = 0, then the above method fails to produce a "reciprocal" nest: division by is here also impossible. The result of the preceding considerations is thus as follows: By definitions 14 to 19, the system of all nests of intervals is ordered in the sense of 4, 1, and admits of having its elements combined by the four processes in the sense of 4, 2. In consequence of the theorems 14 to 19, as stated in each case, this system possesses further, in the aggregate of all rational-valued nests, a sub-system, similar and isomorphous to the system of rational numbers, in the sense of 4, 3. It remains to show that the system also fulfils the Postulate of Eudoxus. But if (x n \ y n ) = a and ( x n I yn) ~ v are an y two positive nests for which all endpoints of in- tervals are positive (cf. 15, 5), let x m and y m f be a definite pair of these endpoints; the theorem of Eudoxus ensures the existence of an integer p, for which p x m > y m ', and the nest p a, or (p x n \ p y n ), in accordance with 15, is then effectually > a'. The next step should be to establish in all detail (cf. 14, 4 and 15, 2) that the four processes defined in 16 to 19 for nests of intervals obey the fundamental laws 2. This again offers not the slightest difficulty and we will accordingly spare ourselves the trouble of setting it forth 26 . The Fundamental Laws of Arithmetic, and thereby the entire body of rules valid in calculations with rational numbers, effectually retain their validity in the new system. By this, our nests of intervals have finally proved themselves in every respect to be numbers in the sense of 4: The system of all nests of intervals is a number-system, the nests themselves are numbers 27 . 26 As regards addition, for instance, it should be shown that: a) Addition can always be carried out. (This follows at once from the defini- tion.) b) The result is unique; i. e. a a', T = T' (in the sense of 14) imply a -f- r a 1 \- r' , if the sums are formed in accordance with 16 and the test for equality carried out in accordance with 14. In the corresponding sense, it should be shown further that c) a + T = T -f- a always. d) fe + a) + T = g -|- (o- + T) always. e) a < a' implies a -\- T < a' 4* T always. And similarly for the other three processes of combination. 27 Whether, as above, we regard nests of intervals as themselves numbers, or imagine some hypothetical entity introduced, which belongs to all the intervals J n (cf. 15, 3) and thus appears to be in a special sense the number enclosed by the nest of intervals and, consequently, the common element in all equal nests this at bottom is a pure matter of taste and makes no essential difference. The equality a -- (x n \ y n ) we may, at any rate, from now on, (cf. 13, footnote 19) read indifferently either as "a is an abbreviated notation for the nest of intervals (x n \ y n )" 9 or as "a is the number defined by the nest of intervals (x n \ y n )". 4. Completeness and uniqueness of the system of real numbers. 33 This system we shall henceforth designate as the system of real numbers. It is an extension of the system of rational numbers, in the sense in which the expression was used on p. 11, since there are not only rational- valued nests but also others besides. This system of real numbers is in one-one correspondence with the whole aggregate of points of the number-axis. For, on the strength of the considerations set forth on pp. 24, 25, we can immediately assert that to every nest of intervals a corresponds one and only one point, namely that common to all the intervals / n , which on account of the Cantor- Dedekind axiom is considered in each case as existing. Also two nests of intervals a and cr' have, corresponding to them, one and the same point, if and only if they are equal, in the sense of 14. To each number cr (that is to say, to all nests of intervals equal to each other) corresponds exactly one point, and to each point exactly one number. The point corresponding in this manner to a particular number is called its image (or representative) point, and we may now assert that the system of real numbers can be uniquely and reversibly represented by the points of a straight line. 4. Completeness and uniqueness of the system of real numbers. Two last doubts remain to be dispelled 28 : Our starting point in 3 was the fact that the system of rational numbers, by reason of its "gaps", could not satisfy all demands which would appear in the course of the elementary processes of calculation. Our newly created number- system the system Z as we will call it for brevity is in this respect certainly more efficient. E. g. it contains 29 a number a for which cr 2 2. Yet the possibility is not excluded that the new system may still show gaps like the old, or that in some other way it may be susceptible of still further extension. Accordingly, we raise the following question: Is it conceivable that a system Z, recognizable as a number-system in the sense of 4, and con- taining all the elements of the system Z, should also contain additional elements distinct from these? * 28 Cf. the closing words of the Introduction (p. 2). 29 For if CT = (x n | y n ) denote the nest of intervals constructed on p. 20 in connection with the equation A? 3 = 2, then by 18 we have a a (x n 2 \ y n *). Since, however, # n 2 < 2 and y n 2 > 2, it follows that a 2 = 2. Q. E. D. 80 I. e. Z would have to represent an extension of Z in the same sense as Z itself represents an extension of the system of rational numbers. 34 Chapter I. Principles of the theory of real numbers. It is not difficult to sec that this cannot be so, so that we have in fact the following theorem: 20. Theorem of completeness. The system /, of all real numbers is in- capable of further extension compatible with the conditions 4. Proof: Let Z be a system which satisfies the conditions 4 and contains all the elements of /. If a denote an arbitrary element of Z, then 4, 4 in which we choose for ft the number 1, contained in Z, and also, therefore, in Z shows that there exists an integer p > a, and similarly another p' > a. For these 3l we have p' < a < p. Considering successively the (finite number of) integers between p' and />, starting with - />', we know that we must come to a last one which is still ^ a. If this be called g, then By applying to this interval g . . . g + 1 the method, already re- peatedly used, of subdivision into ten parts, a perfectly definite nest of intervals (x n \ y n ) is obtained. And a repetition word for word of the proof in 15, 3 shows that the number thus defined can neither be > nor < a. Every element of / is therefore equal to a real number, so that Z can contain no elements other than real numbers. A final objection might be this: We have succeeded in forming the system Z in a comparatively natural, but after all an arbitrary, manner. Other measures, obviously, might be adopted for filling up the gaps in the system of rational numbers. (In the very next section we shall come across other, equally ready means to this end.) It is conceivable that a different method would lead to other numbers, i. e. to number-systems differing, in more or less essential particulars, from the one constructed by us. The question thus indicated may be given a precise formulation as follows: Let us suppose that we have somehow, starting with the system of rational numbers, succeeded in constructing a system < of elements which, besides still satisfying the conditions 4, as is the case with our system Z, and therefore deserving the name of a number-system, also fulfils a further requirement, usually referred to as the Postulate of completeness, on account of the theorem proved above. On the strength of 4, 3, ^ contains elements, corresponding to the rational numbers. Let (x n | y n ) be any nest and let \ n and n be the elements of associated with x n , y n in accordance with 4, 3; the stipulation then runs thus: shall always contain at least one element # satisfying, for every n y the con- ditions r n ^ cs ^ *) n . In exact form, our problem is now: Can such a system <5 differ in [ At this point, the Postulate of Eudoxus gains its axiomatic significance. 4. Completeness and uniqueness of the system of real numbers. 35 any essential particulars from the system Z of real numbers, or must the two systems be regarded as substantially identical, in the perfectly definite sense that they can be brought into relation as similar and isomorphous to one another? The theorem stated below, by solving this problem in the sense which we should anticipate, closes the construction of the system of real numbers. Theorem of Uniqueness. Every such system & is necessarily similar 21. and isomorphous to the system Z of real numbers as constructed by us. Essen- tially, only one such system therefore exists. Proof. By 4, 3, contains a sub-system <', which is similar and isomorphous to the system of rational numbers contained in Z, and whose elements may therefore be called, for short, the rational elements of ^ If a (x n \ y n ) is any real number, 5 rnust, according to our new stipula- tion, contain an element a, which for every n satisfies the conditions in ?? * ^ Wn if \ n and \j n are the elements of S corresponding to the rational numbers x n and y n . Also, these conditions define g uniquely. For if a second element /, simultaneously with *, satisfied the conditions \ n ^ $ <* \^ n for every , then it would follow, word for word as in the proof of 12, that for every n i. e. ^ the non-negative one of the two elements & $' and $' . Let r stand for an arbitrary positive rational number, and i for the cor- responding element in > (therefore in 5'); then, on account of the similarity and isomorphism of >' with the system of rational numbers, we must have, simultaneously with y p x p < r, the relation \j p r^ < v holding for a suitable index p. For every such r therefore If therefore tj denotes one particular such i and if r n , n = 1, 2, . . . , denotes the element (certainly present in >', by 4, 2) which, when repeated n times, yields the sum r lf we see, after writing down the above inequality for r = v n and adding it to itself n times, that for every n = 1, 2, . . . , n | d $' | ^ t! must also hold. Since, however, satisfies the postulate 4, 4, it follows that = *'. If we proceed to associate this uniquely defined clement g and the real number cr, it becomes clear that contains a sub-system 5* $ similar and isomorphous to the system /, of all real numbers. That such a system 5* is n t susceptible of further extension compatible 36 Chapter I. Principles of the theory of real numbers. with the conditions 4, but must be identical with c ij), was the import of the previously established theorem of completeness. Thereby, it is proved that 5 an d %, arc similar and isomorphous to one another, and therefore may be regarded, in all essentials, as identical: Our system Z of all real numbers is in all essentials the only one possible satisfying both the conditions 4 and the postulate of completeness. After these somewhat abstract considerations, the main result of our whole investigation may be summarised as follows: Besides the rational numbers with which we are familiar, there exist others, the so-called irrational numbers. Each of them may be enclosed (determined, given, . . .) by a suitable nest of intervals and this indeed in many ways. These irrational numbers fit in consistently with the rational numbers, in such a manner that the conditions stated in 4 are fulfilled by the joint system of all rational and irrational numbers, with which, to be brief, all calculations may be effected, formally^ exactly as with the rational numbers alone, but with greater success. This wider system is moreover incapable of any further extension compatible with conditions 4, and is in all essentials the only system of symbols which satisfies these conditions 4 and also the postulate of com- pleteness. We call it the system of real numbers. It is with the elements of this system, with the real numbers \ that we work (at first exclusively) in the sequel. We consider a particular real number as given (known, determined, defined, calculable, . . .) if either it is a rational number and so can be literally written down with the help of integers inserting if need be a fractional bar or a minus sign or (and this holds in any case) we are given 32 a nest of intervals defining the number. We shall very soon see, however, that many other ways and means, besides the nests of intervals, exist, for defining a real number. In pro- portion as such ways become known to us, we shall widen the above- mentioned conditions, under which we consider a number as given. 32 I. e. by the complete explicit specification of the (rational) endpomts in the manner just described* 5. Radix fractions and the Dedekind section. 37 5. Radix fractions and the Dedekind section. A few of the methods for defining real numbers may be mentioned at once, as particularly important from the points of view of both theory and practice. In the first place, a nest of intervals need not always be given in the form (x n \ y n ) considered by us ; it may often be written in a more convenient form. Thus, as we have already seen, a decimal fraction, e.g. 1-41421 . . . , may be immediately interpreted as a nest of intervals, with the assumptions 1 =l-4; #a=l-41; ar 3 = 1-414; ..., and, generally, x n equal to the decimal fraction broken off after the if" 1 digit; y n being derived from x n by raising the last digit by one, i.e. y n -- x n -f- 1( y w - Practically, we may thus say that decimal fractions represent a peculiarly clear and convenient specification of nests of intervals 33 . It is obviously quite an unessential part that the base or radix 10 of the ordinary scale of notation plays in this connection. If g is any integer ^ 2, we have the exact analogue for fractions in a scale of radix g or radix fractions with base g. To begin with, given a real number o-, an integer p (>, =, or < 0) is uniquely defined by the condition p^cr <p |-1. The interval y o between p and p -f- 1 is next divided into g equal parts, and each of these parts considered both hero and similarly in the following steps as including its left endpoint, but not its right one. Then cr belongs to one, and to one only, of these parts, i. e. among the numbers 0, 1, 2, . . . , g I there is one and only one which we shall call for brevity a "digit" and denote by #! for which 33 The drawback to it is that we can seldom perceive the law of succession of the digits, i. e the law of formal ion of the .v w 's and >' n 's. 38 Chapter I. Principles of the theory of real numbers. The interval / x thus defined we proceed to divide again into g equal parts, and a will, as before, belong to one, and to one only, of these parts, i. e. a definite "digit" x 2 will be found for which The interval / 2 thus defined we proceed to divide again into g equal parts, and so on. The nest of intervals (/ n ) = (x n \y n ) determined by this pro- cess, for which * 4- f 4- 4- --"=* 4- g + g * + --- + gn - l + z n (n = 1, 2, 3, . . .) clearly defines the number cr, so that M a (# | jy w ). But on the analogy of decimal fractions we may now write o ---- p I O-*! . where of course the base g of the radix fraction must be known from the context. We have therefore the 22. Theorem 1. Every real number can be represented in one and essen- tially only one 35 way by a radix fraction in the scale of base g. We mention the following theorem relating further to this represen- tation, but shall make no use of it in the sequel: Theorem 2. The radix fraction for a real number a whatever be 31 That we have a nest of intervals is immediately obvious, since x n _ 1 < X n <*" y n ^ y n _ 1 throughout, and y n v n - n forms a null sequence, by 10, 7. r> The slight alteration in our method, required if all the intervals are con- sidered as including their right and not their left endpomts, the reader will doubtless be able to carry out for himself. The two results differ if, and only if, the given number a is rational, and can be written as a fraction having, as denominator, a power of g t so that the point a is an endpoint of one of our intervals. Actually the two nests of intervals p -f 0-afi ar, . . . * r _i (~r ~ J ) (g - 1) (# - 1) and / -I- O^ ar t . . . *,._, z r 00 . . . , where the digit z r is supposed ^ 1, are equal by 14. In every other case, two radix fractions which are not identical are unequal, by 14. The reader will easily prove for himself that, except m this case, the representation of any real number a as a radix fraction with base g is absolutely unique. 5. Radix fractions and the Dedekind section. 39 the chosen radix g 2g 2 will prove periodic (or recurring) if and only if a is rational**. A particularly advantageous choice to make is often g = 2 ; the pro- cess for expressing the number a is then called briefly the method of bisection and the resulting radix fraction, whose digits can in that case only be or 1, is called a binary fraction. The method, in a somewhat more general light, is this: we start from a definite interval / and, in accordance with some particular rule or point of view, definitely select one of its two halves, calling it J\ we then again make a definite choice of one of the two halves of y lf calling it / 2 ; and so on. By so doing, we specify, in every case, a well-defined real number, determined with ab- solute uniqueness by the method which regulates at each stage the choice between the two half- intervals 37 . In radix fractions, just as in decimal fractions, we accordingly see a peculiarly clear and convenient mode of specifying nests of intervals. They shall accordingly in future be admitted for the definition of real numbers on the same footing as decimal fractions. The distinction lies somewhat deeper between nests of intervals and the following method of definition of real numbers. We suppose given, in any particular way as , two classes of numbers A and B, subject to the following three conditions: 1) Each of the two classes contains at least one number. 2) Every number of the class A is 5^ every number of the class B. 3) If an arbitrary positive (small) number e is prescribed, then two numbers can be so chosen from the two classes, a ', say, from A and b', say, from B, that 39 b' a < e. Then the following theorem, holds : 30 Here for simplicity we regard terminating radix fractions as periodic with period 0. That every rational number can be represented by a recurring decimal fraction was proved by J. Walhs, De Algebra tractatus, p. 3<>4, 1G ( J3. That conversely every irrational number can always, and in one way only, be represented as a non- recurring decimal fraction was first proved generally by O. Stolz (Allgememe Anth- metik I, p. 119, 1885). 37 An example was given in 12, 2. 88 E. g. A contains all rational numbers whose cube is < 5, B all rational numbers whose cube is > 5. 30 \y e sav f or s hort: the numbers of the two classes approach arbitrarily near to one another. In the example of the preceding footnote, we see at once that conditions 1) and 2) are satisfied; that 3) is also satisfied we recognise from the possibility of calculating (by the method of partition into tenth parts, for instance) two decimal fractions ,v n and y n with n places of decimals, differing only by a unit in the last place, and such that x n 3 < 5, y n * > 5; n being so chosen that , ( . n < e. 40 Chapter I. Principles of the theory of real numbers. Theorem 3. There exists one and only one real number a such that for every number a in A and every number b in B the relation a-^v^b is always true. Proof. It is again obvious that no two different numbers cr, <r' with this property can exist. For putting | a a' \ r, we should have > 0, yet b a ^ c for every pair of elements a and b from A and B respectively, contrary to condition 3. There exists then at most one such number a. We find it in the following way: By hypothesis, there is at least one number a l in A and one number b in B. If a = 6 X , then the common value is manifestly the number a which we are in search of. If a l 4= b ly and therefore by 2), a l < b ly then we choose two rational numbers x l f^ a ly and y ^ b l and apply the method of bisection to the interval / l which they deter- mine; we denote the left or right half by / 2 , according as the left half (endpoints included) does or does not still contain a point of the class B. By the same rule we next select one of the halves of / 2 , calling it / 3 , and so on. The intervals / 1? / 2 , . . . , ./, . . . , being obtained by the method of bisection, necessarily form a nest ( A) = (x n I y n ) = * From their mode of formation, they possess moreover the property that no number of B can lie to the left of any of their left endpoints, and no number of A to the right of their right endpoints. But from this it follows at once that the number a enclosed by them is the number required by theorem 3. In fact, if, contrary to the assertion in that theorem, a particular number a of A were > cr, so that a a > 0, then we could choose from the succession of intervals J n a particular one, sa Y /i> -~ X P ypy Wlt h length < a a. Since x v 5g a ^ y p , this would imply y p or <; y 9 x v <. a a, i. e. y 9 < a, whereas, actually, no point of A lies to the right of the right endpoint y p of y p . If on the other hand, in any instance, b < cr, it would similarly follow that for a suitable index q, b < x qy whereas actually no point of B lies to the left of the left endpoint of an interval J q . Hence we must in- variably have a ^ u fg b. Q. E. D. As a special corollary, we have the following theorem, which sup- plements Theorem 12, forming an extension of it to the case when the numbers there occurring are arbitrary real numbers. In the formulation, we anticipate the obvious definitions 23 25 of next paragraph. 5. Radix fractions and the Dedekind section. 41 Theorem 4- If (x n ) is a monotone ascending, and (y n ) a monotone des- cending, sequence of (any) real numbers ; //, further, x n <^ y n for every n, and the differences y n x n d n form a null sequence-, then there is invariably one and only one real number a, such that for every n We then say, as before (cf. Definition 11), that the two given sequences define a nest of intervals (x n \ y n ) and that a is the number which it (uniquely) deter- mines. Proof. If with all the left endpoints x n we constitute a class A, and with all the right endpoints y n a class #, of real numbers, these clearly satisfy conditions 1) to 3) of Theorem 3, from which the correctness of the above statement at once follows. Remarks and Examples?. 1. Instead of 3), it is often more convenient to stipulate that e.g. every rational number should belong either to A or to B (as \\as the case in the example of last footnote). In fact, in that case, since rational numbers arc dense on the number axis, the requirement 3) is fulfilled of itself. To see this, we have only to imagine the \\hole number-axis subduidcd into equal portions of length < e/2. Now consider any one of the portions containing an element from A, and, to the right of it, take another portion containing an element from B , together with these two portions, take the finite number of portions, if any, between them. One of these considered portions must be the first of them to contain an element b from B. Either this particular portion, or the preceding one, will contain an element a from A, and we have b a ^ . 2. It is often still more convenient to divide till real numbers into tsvo classes A and B. In that case of course 3) is, a fortion, also satisfied of itself. 3. If the two classes A and B are given in one of the last-mentioned ways, then we say that a Dedekind section, is made in the domain of either rational or real numbers, as the case may be 10 . The someuhat more general specification of two classes ll involved in our theorem 3 \\i\\ also for brevity be termed a section and denoted by (A \ B). Our theorem 3 can then be stated briefly in the form: A section (A \ B) invariably defines a determinate real number. And its proof consists simply in pointing out that the specification of a section carries with it the speci- fication of a nest of intervals, which furnishes a number a with the properties required. 4. Seeing then that every section immediately provides a definite nest of intervals, we shall henceforth regard sections as permissible means of defining (determining, specifying, . . .) real numbers; also, we now write, if the section (A | B) defines the number a, (A\B) a. 40 Cf. p. 1 , footnote 0. 41 This was given in the above form by A. C\it>elli % Giornale di Matematici, Vol. 35, p. 209, 1897. 42 Chapter I. Principles of the theory of real numbers. 5. The converse is of course equally true and even more easily proved. Given a nest (x n \ y n ) = cr, we can consider all left endpoints x n as forming a class A, and right endpoints a class B, and these two classes evidently furnish a section, which defines the same numher a as the nest itself. A nest can accordingly be regarded as a particular kind of section. (>. By our last remark, the method of sections (for the definition of real numbers) is superior in generality to that of nests. It is also quite as convenient from the intuitional point of v lew. For if we take, say, the section (A \ B) in the somewhat more special form, mentioned in 2, of a section in the domain of real numbers, then what our theorem implies is this. If we imagine all points of the number-axis separated into two classes A and B, thinking e. g. of points of the one class as marked black and those of the other as white; and if, when this is done, (I) there is at least one point of each kind, (2) every black point lies to the left of every white point, and (3) every point on the number-axis is effectually coloured either black or \\hite, then the t\\o classes must come into contact at a perfectly definite place, and to the left of this place all is black, to the right of it all is \\hite. 7. We must take care, however, not to accept the illustration just given as a proof. Had we not already with the help of nests of intervals invented the class of real numbers, our theorem could not be proved at all any more than it could be proved that every nest defines a number. We simply agreed and were amply justified by the result to regard every nest as a number. In exactly the same way we can agree and this is actually the course followed by JR. Dedekmd 42 in his construction of the system of real numbers to regard every section in the domain of rational numbers as a "real number" , and we should then, exactly as in our investigations in 3, only have to examine whether this is permissible; i. e. we should have to make sure whether the totality of all such sections (A \ Z?) forms a number system in the sense of conditions 4 which is not more difficult than the analogous investigations carried out in 3. Henceforward and for the present exclusively real numbers form our working material. We may even, if we please, drop the word "real": For the present, "number" shall invariably mean a real number. Exercises on Chapter I. 1. From the fundamental laws 1 and 2 deduce the most important of the further arithmetical rules, e. g. (a) the product of two negative numbers is positive; (b) a .+ c < b + c invariably implies a < b ; (c) for every a we have a -= ; etc. 2. When in 3, II, 4 are the signs of equality correct? 3. Express the following numbers as binary and as ternary fractions (i. e. in scales of notation of which the bases are respectively 2 and 3) : 1 3 1 1 10 2' 8' TV 7' 17 ; find the first few figures of the binary and ternary fractions for V2, V3, ir and e. 42 Stetigkeit und irrationale Zahlen, Brunswick 1872, 6. Arbitrary sequences and arbitrary null sequences. 43 a n __ an 4. In the sequence 6, 7 prove x n o t where a and ft are the roots of the quadratic equation x 2 x -f- 1. (Hint: the sequences (a n ) and ()3 W ) have the same law of formation as the sequence 6, 7.) 5. Form the sequence (v n ) of numbers given, for \: 1, by the formula .v nfl ---= ax n -| A \ n _,, where a and A are given positive numbers and the initial terms #, x l 0, 1 ; 1, 0; -- 1, a; 1, j3; or are arbitrary. (Here a and j3 denote respectively the positive and the negative root of the equation x 2 a x -}- b ) In each of the four cases give an explicit formula for x n . 6. If / , /!, / 2 , ... is a sequence of nested intervals (i. e. each contained in the preceding) about whose lengths nothing further is known, then there is at least one point which belongs to all the / n 's. 7. A real number or is irrational, if we can find an ascending sequence of integers (<y n ), such that q n a is not an integer for any H, but if, \\hcn p n stands for the integer nearest to q n a, ( f fn a Pn) 1S a null sequence. 8. Prove that (v n | y n ) is a nest in each of the following examples: n , ..... b) < A:, < 3-, and for every n ^ 1, v nfl *'r n y n , v n H -- } (V M -f- .v n ); c) < x, <- v, , v nM - i (v w f Vn). y n +i ~ VY M .v w : d) ^ Xt <>'! ,>' wf i - 1 (x n i V n ), V/M r v/ ^ v n 3' w 1 1 J e) < .Y! < >'! ,. , A W -H - ^v n .v n , 3'n+i = 2 (v nf i + y n ); g) < v, < Vi ,3'nM i (V w h 3' w ), V, H _, -- Vw '' Vw . 3'w+i Evaluate the numbers defined in examples (a) and (g). (Cf. problems 91 and 92.) Chapter II. Sequences of real numbers. 6. Arbitrary sequences and arbitrary null sequences. We now resume our considerations of 2, and generalise them by allowing all the numbers which there occur to be arbitrary real numbers. Since, with these, we may operate precisely as with rational numbers, both the definitions and the theorems of 2 will, in all essentials, remain unchanged. We may accordingly be brief. 44 Chapter II. Sequences of real numbers. 23. Definition 1 . If to each positive integer 1, 2, 3,..., corresponds a definite real number A:,,, then the numbers are said to form a sequence. Examples 6, 1 12, may, of course, also serve here. Similarly, the Remarks 7, 10 retain full validity. We give a few more examples, in which it is not im- mediately apparent whether the numbers in question are rational or not. Examples. 1. Let a 0,:W10 . . . , i. e. equal to the decimal fraction whose first few digits were obtained in a footnote (p. 24) from the equation 10 l 2; and put x n -~= a n for n =-- 1, 2, ;}, . . . 2. With the same meaning for a, let x n = 3. Apply the method of .successive bisection to the interval / ... 1, taking first the left half, then twice running the right half, then for the next three steps again the left half, then four times running the right half, and so on. Denote the number 2 so defined by 6 (\\h\t is its value, approxim itely 5 ), and put foi x n , successively, +*,-*,+;, -j, +*, -*, +i. -;, -i *.... 4. With the same meaning for b, put for x n , successively, 1 - b, 1 + b, 1 - b*, 1 | b\ 1 - b\ 1 + 6 3 , . . . 5. W T ith the same meaning for a and 6, let v t b'j the middle point of the stretch between them, i. e. x l J (a f 6); x z the middle point between je t and 6, # 3 , that between * 2 and a, x 4 , that between #, and b; i. e. generally, x n+l , the middle point between x n and either a t or b y according as n is even or odd. Definitions: 1. A sequence (x n ) is said to be bounded if a constant K exists, such that the inequality is satisfied for every n. 2. A sequence (x n ) is said to be monotone increasing if x n 5^ x n+l for every n\ monotone decreasing, if x n ^ x n+1 for every n. All remarks made in 8 and 9 retain their full validity. 1 For the meaning of the mark cf. the preface, as also later the beginning of 52. 2 Written as a binary fraction, 6 ^ 01 10001 1 1 10 ... 6. Arbitrary sequences and arbitrary nyjl sequences. 45 Examples. 1. The sequences 23, 1, 2, 4 and 5 are evidently bounded. Sequence 3 is not bounded, and in fact neither on the left nor on the right; for we certainly have < 6 < - and therefore , ^ > 2 m > m, and accordingly ,- < m. Terms of the sequence may therefore always be found, which are > K or < K y how- ever large the constant K is chosen. For 5, the boundedness follows from the fact that all the terms lie between a and b. 2. The sequences 23, 1 and 2 are monotone decreasing: the others are not monotone. The definition 10 of a null sequence and the appended remarks which the student should read through again carefully also remain unchanged. Definition. A sequence (x n ) shall be termed a null sequence if, 25. subsequently to the choice of an arbitrary positive numbers, a number n Q = n (e) may always be assigned, such that the inequality 3 is fulfilled for every n > . Examples. 1. The sequence 23, 1 is a null sequence, for the proof 10, 7 is valid for any real a, for which | a \ < 1. 2. 23, 2 is also a null sequence, for here | x n \ < , therefore < e, provided . 1 n n > . e For null sequences these will later on play a dominating part a number of quite simple theorems, which will be continually applied in the sequel, will also be proved here. The following two, in the first place, are obvious enough: Theorem 1. If (x n ) is a null sequence and the terms of the sequence 26. ( x n')> f or w er y n beyond a certain value m, satisfy the condition \ x n f \ ^ | x n \ 9 or, more generally, the condition \ Xn '\^K-\x n \, in which K is an arbitrary (fixed) positive number, then x n ' is also a null sequence. (Comparison test.) 3 Given any positive real number e, a positive rational number e' < e can be designated; in fact, by the fundamental law 2, VI, we can find a natural number n > , and e' satisfies the requirements. From this it follows that, for rational sequences, the above definition is equivalent to the definition 10, in spite of the. fact that only rational e were allowed there. 46 Chapter II. Sequences of real numbers. Proof. If the condition | x n ' \ ^ K \ x n \ is satisfied for n > m and e > is given, then by the assumptions we can assign H O > m, so that for every n > n Q , | x n \ < . Since for these values of n we then /C also have | x n ' \ < , (x n f ) is therefore a null sequence. The following theorem is only a special case of the preceding: Theorem 2. If (x n ) is a null sequence, and (a n ) any bounded sequence, then the numbers x n f = a n x n also form a null sequence. On account of this theorem we say for short: A null sequence "may 1 * be multiplied by a bounded factor. Examples. 1. If (x n ) is a null sequence, io* lf fa 10*3, fa 10 * 6 ... is also a null sequence. 2. If (x n ) is a null sequence, so is (| x n |). 3. A sequence, all of whose terms have the same value, say c, is certainly hounded. If (x n ) is a null sequence, (c x n ) is therefore also a null sequence. In particular, f-J, (c a n ) for | a \ < 1, etc. are null sequences. The next propositions are less obvious: 27* Theorem 1. If (x n ) is a null sequence, then every sub-sequence (x n f ) of (x n ) is a null sequence 4 . Proof. If, for every n > n , | x n \ < z, then we have, ipso facto, for any such n, since k n is certainly > , when n is. Theorem 2. Let an arbitrary sequence (x n ) be separated into two sub-sequences (x n f ) and (x n "), so that, therefore, every term of (x n ) belongs to one and only one of these sub-sequences. If (x n ') and (x n ") are both null sequences, then so is (x n ) itself. 4 If ki < k 2 < k 3 < . . . < k n < . . . is any sequence of positive integers, then the numbers x n ' = x kn (n = 1, 2, 3, . . .) are said to form a sub-sequence of the given sequence. 6. Arbitrary sequences and arbitrary null sequences. 47 Proof. If a number e > be chosen, then by hypothesis a num- ber n exists, such that for every n > n, |# n '| < <e, and also a num- ber n" y such that for every n > w", | #"!< ^ nc terms # n ' with index <^ n' and the terms # n " with index <Ln", have definite places, i. e. definite indices, in the original sequence (x n ) . If n Q is the higher of these indices, then for every n > M O , obviously | x n \ < e, q. e. d. Theorem 3. // (# n ) is a null sequence and (x n ') an arbitrary rearrangement* of it, then (x n ') is also a null sequence. Proof. For every n > w , | x n \ < e. Among the indices belong- ing to the finite number of places which the terms x lt # a , . . ., x n occupy in the sequence (# n '), let ri be the largest. Then obviously, for every n > n', |# n '| < e; hence (x n f ) is also a null sequence. Theorem 4. // (x n ] is a null sequence and (x n ') is obtained from it by any finite number of alterations*, then (x n f ) is also a null se- quence t. The proof follows immediately from the fact, that for a suitable integer />^0, from some n onwards we must have x n ' = x n+ . For if every x n for n ^> n l has remained unchanged, and #, ?1 has received the index n f in the sequence (x n ')> then in point of fact for every n > ri, if we put p = Wj n Theorem 5. // (x n ') and (x n ") are two null sequences and if the sequence (# n ) is so related to them that from a certain m onwards then (x n ] is also a null sequence. Proof Having chosen e > 0, we can chose n >> m so that, for every n > n Qt e < x n ' and a? n " -< + e. For these ris we then have, ipso facto, e < x n < + , that is | x n \ < e', q. e. d. 6 If ki t k 2 , . . , fe n , ... is a sequence of positive integers such that every in- teger occurs once and only once in the sequence, then the sequence formed by is said to be a rearrangement of the given sequence. 6 We will describe this concept as follows: If we alter any sequence, by omitting, or inserting, or changing, a finite number of terms (or by doing all three things at once), and then renumber the altered sequence, \vithout changing the order of the terms left untouched, so as to exhibit it as a sequence Cv rt '), then >\e shall say, (x n ') is obtained or has resulted from (x n ) by a finite number of alterations. 7 It is precisely because of this theorem that one may say of a sequence that the property of being a null sequence concerns only the ultimate behaviour of its terms (cf. p. 10). 48 Chapter 11. Sequences ot real numbers. Calculations with null sequences, finally, are founded on the following theorems: 88. Theorem 1. // (x n ) and (x n ') are two null sequences, then i. e. the sequence whose terms are the numbers y n = x n -f- x n ', is also a null sequence. Briefly. Two mill sequences "may" be added term by term. Proof. Ife>0 has been chosen arbitrarily, then by hypothesis (cf. 10, 4 and 12) a number n 1 and a number w 2 exist such that for every n > !, | x n | < ?, and for every n > 2 , | x n ' \ < . If w is a number 2i & ? //! and 2g 2 then for n > n Q I y n \ = I *. + *.' I = I * I + 1 *' I < I + g = * (y n ) is therefore a null sequence 8 . Since, by 26, 3 (or 10, 5), ( a?/) is a null sequence if fa^') is, (y n ') = (x n as n ') is then by the above also a null sequence, i. e. we have the theorem- Theorem 2. // (a? n ) and (x n ') are null sequences, then so is (y n ') == (x n x n '). Or briefly: null sequences "may" be subtracted term by term Remarks. 1. Since we may add two null sequences term by term, we may also do so with three or any definite number of null sequences. For supposing this prov- ed for (p I) null sequences (a^ 1 ), (#")> > (&%* ~ X) ) , i. e. supposing the sequence to be already recognised as a null sequence, Theorem 1 ensures that the sequence (x n ), for which is also a null sequence. The theorem thus holds for every fixed number of null sequences. 2. That two null sequences "may" also be multiplied term by term, is immediately clear from 26, 1, since null sequences, by 10, 11, are necessarily bounded. 3. Term by term division t on the contrary, is in general not allowed, as is already obvious, for instance, from the fact that when x n =}= 0, is con- Xn 11 x stantly = 1 . If we take x H *= 9 xJ = = , then the ratios ~ do not even pro n n* x m f vide a bounded sequence. 8 For the last inequality 3, II, 4 is used. 7. Powers, roots and logarithms. Special null sequences. 49 4. In the case of other sequences (x n ) also, little can be said in the first instance about the sequence ( ) of the reciprocal values. The following is \x n / an obvious, but often useful theorem: o Theorem 3. // the sequence (\ x n \) of absolute values of the terms of (x n > have a positive lower bound, if, therefore, a number y > exists, such that for every n, then the sequence { ) of reciprocal values is bounded. \x n J In fact, from | x n | > y > it at once follows that for K = we have <K x n for every n. In order to increase the scope both of the application of our con- cepts and of the construction and solution of examples, we insert P. paragraph on powers, roots, logarithms and circular functions. 7. Powers, roots and logarithms. Special null sequences. As, in the discussion of the system of real numbers, it was not our intention to give an exhaustive treatment of all details, but lather to put fundamental ideas alone in a clear light, assuming as known, thereafter, the body of arithmetical rules and concepts, with which after all everyone is thoroughly conversant, so here, in the discussion of powers, roots and logarithms, we will restrict ourselves to an exact elucidation of the basic facts, and then assume known the details of their application. I. Powers with integral exponents. If x is an arbitrary number, we know that the symbol x k for positive integral exponents k ^ 2 is defined as the product of k factors, all equal to x. Here we have therefore only another notation for something we know already. By x 1 we mean the number x itself, and if x =}= 0, it is convenient to agree, besides, that x represents the number 1, x~* the number -^ (k = 1, 2, 3, ...y, so that x 9 is defined for every integral p^O. For these po \\crs* with integral exponents, we merely emphasize the following facts: 1. For arbitrary integral exponents p and q ($0) the three 29. fundamental rules hold: * x p is a power of base x and exponent p. This continental use of the word power cannot be here dispensed with, in spite of the slight ambiguity resulting- from by far the most frequent use of the word in English to designate the exponent. This sense should be entirely discarded from the reader's mind, notably for 35, 2 a and others. (Tr.) 50 Chapter TI. Sequences of real numbers. from which all further rules may be deduced, which regulate calcu- lations with powers . 2. Since, in a power with integral exponent, merely a repeated multiplication or division is involved, its calculation has of course to be effected by 18 and 19. If therefore x is positive and defined for instance by the nest (x n \y n ), with all its endpoints ^> (cf. 15, 5), then we have simultaneously with -fcJyJ' x * = ( xk n \y k n) at once > for all positive integral exponents: and similarly with appropriate restrictions for x <^ or k <I o . 3. For a positive x we have furthermore according as xl as we at once deduce from #^1, if we multiply (v. 3, I, 3) by x". And quite as simply we find: If x^ y a; a and the integral exponent k are positive, then x*^x according as a^^Sg. 4. For positive integral exponents n and arbitrary a and 6 we have the formula + (n\ n-lt -ik i I fn\ jn (*)* b H t-y*. where [? ) , for l^Lk^n, has the meaning R fn\ _ n (n - 1) (n - 2) . . . (n - fe-f 1) W 1 - 2 3 ... k and (Q] will be put=l. (Binomial Theorem.) II. Roots. If a be any positive real number, and k a positive integer, then shall denote a number whose & th power = a . What interests us here is solely the existence question: Is there such a number, and to what extent is it determined by the problem thus set? This is dealt with in the 9 In this, the value for the base x or y is only admissible if the cor responding exponent is positive. 7. Powers, roots and logarithms. Special null sequences. 51 Theorem. There is, invariably, one and only one positive number f 30. satisfying the equation ft = * (a > o) IP - - We write g = y and call { the & th root of a. Proof. One such number may immediately be determined by a nest of intervals, and its existence thereby established We use the decimal-section method. Since 0* = < a, but, p denoting any positive integer > a, p 16 ^ p > a , there is one and only one integer g ^> for which 10 k . , , . fc g ^<(g+1) I he interval / determined by g and (g -f- 1) we divide into 10 equal parts and obtain, in the manner now repeatedly worked out, a defi- nite one of the digits 0, 1, 2, . .., 9, which we may denote, say, by z v and for which and so on, and so on. We therefore obtain a nest of intervals (^J = (x n | y^ whose endpoints have definite values of the form and v a -L * -L- **--] _____ L **-_ J_ ? 1 y n 6 i jo ^ 10 2 r ~r 10 ,,_i r 10 If f = (aj^ | yj be the number thereby determined, then since here all endpoints of intervals are ^> 0, it at once follows by 29, 2 that But, by construction, x k <^a <^y k for every n, hence, by 5, Theorem 4, we must have ,. * = *. That this number f is, moreover, the only positive solution of the problem, follows directly from 29, 3, since it was there pointed out that for a positive ^ ={=, necessarily f* 4 s *> i e. 4=^- If & is an even number, then is also a solution of the problem. We shall not, however, take this into account in the follow- ing pages, but interpret the th root of a positive number a as meaning only the positive number f, completely and uniquely deter- mined by 30 11 . For a = 0, we may also put Va = 19 10 g is the last of the numbers 0, 1, 2, ..., p whose &th power is <. 11 In accordance with this we have, for instance, ^ x* not always =#, but always = | x \ . fc 19 For negative a's we will not define y a at all; we can, however, if h is odd, write |/T= 52 Chapter II. Sequences of real numbers. We will not enter further into the rules for calculations with roots, but consider them as familiar to every one, and will only prove the following simple theorems: 29, 3 gives at once the * ^ * il. Theorem 1. // a > and a x > 0, then V a ^ Va l , according as a ^ a . Further we have the Theorem 2. If a > 0, then \V a) is a monotone sequence; and we have, more precisely, / 3 /~- a>\a>V a >>!, if a >1, to* r- 3 / a<v/a<Vfl <<!, ^f a <l. (For a = 1 , //* sequence ^s of course = 1 .) Proof By 29, 3, a> 1 involves a 71 ^ 1 > a n >1, and thereiore by the preceding theorem, taking w (w -j- l) th roots , n n-H Va > l/a>l. Since for a < 1 aW the ir equality signs are reversed, this proves the whole statement. Hence finally we deduce the Theorem 3. If a > , then the numbers x n = # a" 1 /om a nw/J sequence (monotone by the preceding theorem). Proof. For a=l, the assertion is trivial, as then x n = Q. If n. a > 1, and therefore Va > 1, i. e. x n = Va 1 > 0, then we reason n as follows: By the inequality of Bernoulli (v. 10, 7), Va = 1 + B glVCS a = (l+*J">l + na n >***.. Consequently a; n = | n | < ~, therefore (xj, by 26, 1 or 2, is a null sequence. If 0<a<l, then >1, and so, by the le^ult obtained, a is a null sequence. If we multiply this term by term by the factors Va, n which certainly form a bounded sequence, as a < V a < 1 , then it at once follows, by 26, 2, that (l Vaj, and therefore also (# n ), as a null sequence, q. e. d. 7. Powers, roots and logarithms. Special null sequences. 53 III. Powers with rational exponents. We again regard as substantially known, in what manner one may pass from roots with integral exponents to powers with any rational - > exponent: By a q , with integral p~Q, q > 0, we mean, for any posi- tive a, the positive number uniquely defined by 1L If p > 0, then a may also be == 0; a q must then be taken to have the value 0. With these definitions, the three fundamental rules 29, 1, i.e. the formulae a? a r> = a r + r '; a r b r = (a b) r ; (a r Y = a rr> remain unaltered, for any rational exponents, and therefore calculations \vilh these powers are formally the same as when the exponents are integers. These formulae contain, at the same time, all the rules for working with roots, since every root may now be written as a power with a rational exponent. Of the less known results we may prove, as they are particularly important for the sequel, these theorems: Theorem 1. When a > 1 , then a r > 1 , if, and only if, r > . 32. Similarly, when a < 1 (but positive), then a r is < 1 if, and only if, r>0. Proof. By 31, 2, a and V 'a are either both greater or both less than 1; by 29 the same is true of a and \V a) = a r if and only if p> 0. Theorem la. // the rational number r > 0, and both bases are positive, then a r ^a 1 r , according as a a^. The proof is at once obtained from 31, 1 and 29, 3. Theorem 2. If a > 0, and the rational number r lies between the rational numbers r' and r", then a r also always lies between a r ' and a r " 13 , and conversely, whether a be <, =or>l, and /<, = or >r". Proof. If, firstly, a > 1 and r' < r", then 13 The term "between" may be taken, as we please, either to include or exclude equality on both sides, excepting when a = 1, and therefore all the powers a r also = 1. 3 (061) 54 Chapter II. Sequences of real numbers. By Theorem 1, this already proves the validity of our statement for this case, and in the other possible cases the proof is quite as easy. From this proof we deduce, indeed, more precisely, the Theorem 2 a. If a>l 9 then to the larger (rational) exponent also corresponds the larger value of the power. If a < 1 (but positive) then the larger exponent gives the smaller power. In particular: If the (positive) base a=%=! 9 then different exponents give different powers. Hence we deduce, further, Theorem 3. // (r n ) is any (rational) null sequence, then the numbers x n =* n -l, (fl>0) also form a null sequence. If (r^ is monotone, then so is (# n ). Proof. By 31, 3, \ty~a ij and \y - l) are null sequences. If therefore e > be given, we can so choose M A and n % that ! n for n>n 1 , \Va 1 I n /l and for n > n.,, I V 1 - | * a If m is an integer larger than both n l and n 9 then the numbers \a m I/ and \a m I/ both lie between e and -|-g, i. e i i a m and a m lie between 1 e and 1 + e By Theorem 2, a r then lies between the same bounds, if r lies be- 1 . , 1 tween and -| . Wl Wl that for every n > n , tween -- and -I -- . By hypothesis we can, however, so choose # , '. or -<^< ; for w>n , r<> is therefore between 1 e and 1-J-e. Hence, for these w's, . I a n 1 I < e, proving that (a r 1) is a null sequence. That it is monotone, if (r n ) is, follows immediately from Theorem 2 a. These theorems form the basis for the definition of IV. Powers with arbitrary real exponents. For this we first state the 88. Theorem. // (x n \y n ) is any nest of intervals (with rational end- points) and a is positive, then for a ;> 1, o = (a* n \ a v ) and for a<l> o = (a Vfl \ a* n ) 7. Powers, roots and logarithms. Special null sequences. 55 is also a nest of intervals. And if (x n \y n ) is rational valued and = r, then o = a r . Proof. That in either case the left endpoints form a monotone ascending sequence, the right endpoints a monotone descending se- quence, follows at once from 32, 2 a. By the same theorem, a* n < a Vn> in the one case (a J> 1) and a Vn < a n in the other (a < 1), for every n. Finally, that in both cases the lengths of the intervals form a null sequence, follows, with the aid of 26, from for here the first factor, by 32, 3, is a null sequence, because (y n x n ) is by hypothesis a null sequence with rational terms; and the second factor is bounded, because for every n < a* n <; a yi in the one case (a 2> 1), <U" in the other (a <jl). Now if (# w |jy w ) = y, then r lies between x n and y n , for every n, and so by 32, 2, a r lies between a* and a Vn , for every n; hence by 5, Theorem 4, necessarily a = a r . In consequence of this theorem, we may agree to the following Definition 14 . If a > 0, and Q = (x n I ^ n ) is an arbitrary real number, then: ' a* n a Vn it a > 1 a* = <7, i. e. if This definition can of course only be regarded as appropriate, if the concept of a general power thereby determined obeys subs tan. tially the same laws as the type of power so far considered, that with rational exponents. That this is so, in the fullest sense, is shewn by the following considerations. 1. For rational exponents, the new definition gives the same result 34. as the old. 2. If e (?', then 15 a? a?'. 14 This combination 33 of theorem and definition is, from the point of view of method, of exactly the same kind as those set forth in 14 19: What is demonstrable in the case of rational exponents is raised, in the case of arbitrary exponents, to the rank of a definition, whose appropriate- ness has then to be verified. 16 This assertion, formally rather trivial in appearance, when put some- what more explicitly, runs thus: If (x n \ y n ) = (> and (x n f \ y^ = (>' are two nests of intervals, which may be regarded as equal in the sense of 14, then so are those nests of intervals equal (again in the sense of 14), which by Definition 33 give the powers a e and a e '. 56 Chapter II. Sequences of real numbers. 3. For two arbitrary real numbers Q and Q', and positive a and 6, the three fundamental rules hold, so that with the general powers now introduced we may cal- culate formally in precisely the same way as with the special types hitherto used. Into the extremely simple proofs of these facts we will, as emphasized on p. 49, not enter further 16 ; we will also, so far as concerns the extension of theorems 32, 1 3 to general powers, now immediately possible, content ourselves with the statement and a few indications of the proof. We have therefore the theorems, generalized from 82, 13: 85. Theorem 1. When a > 1, we have a Q > 1 if, and only if, Q > 0. Similarly, when a <. 1, (but positive), we have a Q < 1 if, and only if, Q>0. For by 82, 1, we have e. g. for a > 1, a* n > 1 if, and only if, x n >0. Theorem la. // the real number Q is > 0, and both bases are positive, then a Q ^ a?, according as a ^ a . Proof by 82, la and 15. Theorem 2. // a > and Q is between Q' and Q", then a^ is al- ways between a&' and ae". The proof is precisely the same as 82, 2. It yields, more exactly, the Theorem 2 a. // a > 1, then to the larger exponent corresponds the larger value of the power \ if a < 1 (but positive), then the larger exponent gives the smaller power. In particular: If a + 1> then different exponents give different powers. And from this theorem, exactly as in 32, 3, follows the final 16 As a model we may sketch the proof of the first of the three fundamental rules: If Q = (x n \ y n ) and Q' = (x n ' \ y n *), then by 16, o -J- Q' = (x n + x n ' \ y n -f y^ and therefore we assume a > 1 : Since all endpoints (as powers with rational exponents) are positive, we have, by 18, a e. a e' = (a Xn -a x * \ a v -a Vn "). Since, however, for rational exponents, the first of the three fundamental rules has already been seen to hold, this last nest of intervals is not only equal, in the sense of 14, to that defining a e+e , but even coincides with it term by term. 7. Powers, roots and logarithms. Special null sequences. 57 Theorem 3. // (p n ) is any null sequence, then the numbers form a null sequence. If (g n ) is monotone, then so is (xj. As a special application, we may mention the Theorem 4. // (# n ) is a null sequence with all its terms positive, then for every positive a, /*. ' _ rf X n ~ X n > is also the term of a mill sequence. Thus ( ) for every a > is a null sequence. ^ n ' i Proof. If s > be given arbitrarily, e a is also a positive number. By hypothesis, we can choose n Q so that, for every n > n Q (cf. 10, 1 and 12), For n > n , by 35, la, we then also have, however, which at once proves the whole statement. The above theorems comprise the main principles used in cal- culations with generalized powers. V. Logarithms. The foundation for the definition of logarithms lies in the Theorem. // a > and b > 1 are two real, and in all further 36. respects quite arbitrary numbers, then one and only one real number f always exists, for which b* = a. Proof. That at most one such number can exist, already follows from 35, 2 a, because the base b with different exponents cannot give the same value a. That such a number does exist, we show con- structively, by assigning a nest of intervals which determines it, thus for instance by the method of decimal sections: Since b > 1, (b~ n ) = fp-J is a null sequence, by 10, 7, and there exists, conse- quently, since a and are positive, natural numbers p and q for which b~ p <a and b" 9 < or b 9 > a. a If, now, we consider the various integers between p and -(- q in succession, as exponents of &, there must be one, and can be only one call it g for which b a a, but 58 Chapter II. Sequences of real numbers. The interval J Q =- g . . . (g -[- 1) thereby determined we divide into 10 equal parts and obtain, just as on p. 51, a "digit" z , for which r , but b'*"* r By repetition of the process of subdivision we find a perfectly definite nest of intervals \ x n == S+ jo + + io-i ' 10*' *-(*.|yJ. w " h L_ g + + ..+>-, + , + i f for which for every n, for which, therefore, in accordance with 33, This theorem justifies us in the following Definition. // a > and b > 1 are arbitrarily given, then the real number f , uniquely determined by b * === # t's called the logarithm of a to the base b; and, symbolically, (g is also called the characteristic, and the set of the digits z l9 z, z >A ... the mantissa, of the logarithm.) We speak of a system of logarithms, when the base b is assum- ed fixed once for all and the logarithms of all possible numbers are taken to this base 6. The suffix b in log & is then usually omitted as superfluous. Very soon a particular real number, usually denoted by e, appears quite naturally as the most convenient for all theo- retical considerations; the system of logarithms built up on this base is usually called the system of natural logarithms. For practical purposes, however, the base 10 is, as we know, the most convenient; logarithms to this base are called common or Briggs' logarithms. These are the logarithms found in all the ordinary tables 17 . The rules for working with logarithms we assume, as we did with powers, to be already known, and content ourselves with a mere mention of the most important of them. If the base b > 1 is arbitrary, 17 As a matter of course, a system of logarithms may also be built up on a positive base less than 1. This, however, is not usual. The first logarithms cal- culated by Napier in 1014 were, however, built up on a base b < 1, which presents some small advantages, particularly for logarithms of trigonometrical functions. Neither Napier nor Briggs, however, really used any base. The idea of logarithms as the inverse of powers only developed in the course of the 18th century. 7. Powers, roots and logarithms. Special null sequences. 59 but assumed fixed in what follows, and if a, a', a" ... denote any positive numbers, then 1. log (a! a") = log 0' + log a". 37, 2. log 1=0; log = log a; log 6 = 1. 3. log a Q = Q log a (Q arbitrary, real). 4. log a^ log 0', according as a 5 a'; in particular, 5. log 0^0, according as 0^1. 6. If b and ^ are two different bases (> l), and and x the logarithms of the same number a to these two bases, i. e. then as follows at once from (a =) b% = fr^ 1 , by taking logarithms on both sides to ihc base b and taking account of 87, 2 and 3 7. ff -)> n = 2, 3, 4, ... is a null sequence. In fact ^ < , provided log w > , that is, n> b e . VI. Circular functions. To introduce the so-called circular functions (the sine of a given angle 18 , with the cosine, tangent, cotangent etc.) in an equally strict manner, i e. avoiding on principle all reference to geometrical in- tuition as element of proof and founding solely on the concept ot the real number, is at this stage not yet possible. This question will be resumed later ( 24). In spite of this, we will unhesitatingly enlist them to enrich our applications and enliven our examples (but of course never to prove general propositions), in so far as their know- ledge may be presupposed from elementary work. Thus e. g. the following two simple facts can at once be ascertained: 37tt. 1. If a, , <x 2 , . . ,, a n , . . . are any angles (that is to say, any numbers), then (sin a n ) and (cos ) are bounded sequences; and 18 Angles will in general be measured in radians If in a circle of radius unity we imagine the radius to turn from a definite initial position, then we measure the angle of turning by the length of the path which the extremity of the moving radius has traversed taking it as positive when the sense of turning is counterclockwise, otherwise as negative. An angle is accordingly a pure number; a straight angle has the measure -J- n or n y a right angle the measure -f- -~- or -,- . To every definitely placed angle there belongs an It a infinite number of measures which, however, differ from one another only by integral multiples of 2jt, i. e. by whole turns. The measure 1 belongs to the angle, the arc corresponding to which is equal to the radius, and which there- fere in degrees is 57 17' 44"-8 nearly. Chapter II. Sequences of real numbers. 2. the sequences and are (by 26) null sequences, for their terms are derived from those of the null sequence f J by multiplication by bounded factors. VII. Special null sequences. As a further application of the concepts now defined, we will examine a number of special sequences: 88. 1. // \a\ < 1, then besides (a n ) even (na n ) is a null sequence. Proof. Our reasoning is analogous to that of 10, 7 19 : For a = 0, the assertion is trivial; for 0<|<z|<l, we may write, with Q > 0, 101 = ,-4--, and therefore \a n \ = 1+c Since here in the denominator each term of the sum is positive, we have for every n > 1, i ni 1 tr i ni 1-2 </-TN > therefore wa < ___. Thus we have \na n \<.e, as soon as - ( '-^. 11 ( n 1)0 i. e. for every The result thus proved is very remarkable: it asserts, in fact, that for a large n the fraction . n is very small, and its denominator therefore very much greater than its numerator. This denominator is however constant (= l) for o = 0, and when Q is very small (and positive), it only increases very slowly with n. Nevertheless, our result shows that provided only n be taken sufficiently large, the deno- minator is very much larger than the numerator 20 . The point % from which | n a n \ = i^r-y lies below a given e we found n = 1 + i does indeed lie very far to the right, not only when e, but also when Q = p 1, is very small (i. e. | a \ very near to l). Substantially this 19 Except that a and Q need no longer be rational. 7. Powers, roots and logarithms. Special null sequences. 61 and only this is true : However | a \ < 1 and e > may be given, we have always, from a readily assignable point onwards, | n a n \ < s. From this result many others may be deduced, e. g. the still more paradoxical fact: 2. // | a | < 1 and a real and arbitrary, then (n* a n ) is also a null sequence. Proof. If a <I 0, then this is evident from 10, 7, because of 26, j_ 2; if a > 0, write | a \ a a l9 so that by 35, la, the positive number a 1 is also < 1. By the preceding result, (n a n ) is a null sequence. By 35, 4 [na^ 1 ]*, i.e. n^ \ a \ n or | n* a n \ , therefore, finally, (by 10, 5), n* a n itself is also the term of a null sequence 21 . 3. If a > 0, then ( /*) is a null sequence 22 , to whatever base b>l the logarithms are taken. Proof. Since b > 1, a > 0, we have (by 35, la), b >1. There- fore (j^n) is a null sequence, by 1. Given > 0, we have consequently from a certain point onwards, say for every n > m n < e' = (ft') n " tP But, in any case, if g denote the characteristic of log n (so that g rg log # < f- 1). If, therefore, we take n > b m , log n, and # fortiori g + 1, is > wz. Hence the last value above, with our choice of w, is < e for every ;/ > ;/ = 6 m . 20 Writing as above | a | = f"T^~ I w a n | -^ ^-^ .> w e may also say: (1 -f- g) n becomes for a positive g more pronouncedly large, or, also more pro- nouncedly infinite, than n itself, by which we again (cf. 7, 3) mean nothing more and nothing less than that our sequence is precisely a null sequence. For future reference we remark here that the results proved in 1 and 2 arc also valid for a complex a, provided only | a \ < 1. 21 With the same change of notation as above, we may say here: "(I + 0) n becomes more pronouncedly infinite than every (fixed) power however large of n itself". 28 Or, in words, "log n becomes less pronouncedly large than every power, how- ever small (but determinate and positive), of n itself". 3 ( G 51 ) Chapter IT. Sequences of real numbers. 4. // cc and f> arc arbitrary positive numbers, then "\ 7 is a null sequence , however large cc and however small ft may be 23 . Proof. By 3., ( fi^ ] is a null sequence, because > 0; by 35, 4, therefore, so is the given sequence. 5. (a? n )=(Vn ij is a mill sequence. (This result is also very remarkable. For when n is large, we have a large number under the V ' the exponent of the V is, it is true, also large; but it is not at all evident a priori which of the two radicand or exponent will, so to speak, prove the stronger.) n Proof. For n > 1, we certainly have \n > 1, therefore n x n = V n 1 certainly ;> 0. Hence in all the terms of the sum are positive. Consequently we have, in particular, n(n 1) 9 " - or 24 .. - 1 _ n Hence _ so that (x n ) = V/w 1 is in fact by 26, 1 and 35, 4 a null sequence. 6. // (#J is a mill sequence whose terms are all > 1, then for every (fixed) integer k, the numbers also form a mill sequence 3 "Every power of log n, however large, (but fixed) becomes less pronouncedly large than every power of n itself, however small (but fixed). 84 The substitution, when n > 1 , of the value n ^- for (n 1) which 6 it cannot exceed, is an artifice often useful in simplifying 1 calculations. a& By the assumption that all a? n 's >> 1, we merely wish to ensure that the numbers x n ' are defined for every n. From a definite point onwards this is automatically the case, since (x n ) is assumed to be a null sequence and therefore from some point certainty | x n \ <[ l f and hence x n > 1. 7. Powers, roots and logarithms. Special null sequences. 63 Proof. From the formulae set forth on p. 22, Footnote 13, where k . we put a = i/1 -f- x n and 6 = 1, it follows that 2fl therefore, since the terms in the denominator are all positive and the last is 1, Irr 'I < I r I x n I ^ I X n > whence, by 26, the statement at once follows. 7. // (x n ) is a null sequence of the same kind as in 6., then the numbers JfH = log (l + .r, t ) also form a null sequence. Proof. If b > 1 is the base to which the logarithms are taken, and e > is given, we write so that we have z l -=- b* 2 > 2 > 0. We then choose n so large, that for every n > w , | x n \ < s 2 . For those w's we have, a fortiori, therefore (by 35, 2 or 37, 4) with which the statement is proved. 8. // (x n ) is again a null sequence of the same kind as in 6., then Hie numbers also form a null sequence, if Q denote any real number. Proof. By 7. and 26, 3, the numbers form a null sequence. By 35, 3 and 37, 3 the same is true of the numbers 6*"-l = (l + sJ*--l = * n , q-e.d. We assume ft >: 2, since for k = 1 the assertion is trivial. 64 Chapter II. Sequences of real numbers. 8. Convergent sequences. Definitions. So far, when considering the behaviour of a given sequence, we have been chiefly concerned to discover whether it was a null sequence or not. By extending this point of view somewhat, in a manner which readily suggests itself, we reach the most important concept of all with which we shall have to deal, namely, that of the convergence of a sequence. We have already (cf. 10, 10) described the property which a sequence (x n ) may have, of being a null sequence, by saying that its members become small, become arbitrarily small, with increasing n. We may also say: Its terms, as n increases, approach the value 0, without, in general, ever reaching it, it is true; but they approach arbitrarily near to this value in the sense that the values of its terms (that is to say, their differences from 0) sink below every number e (> 0), however small. If we substitute for the value in this conception any other real number , we shall be concerned with a sequence (x n ) for which the differences of the various terms from the definite number that is to say, by 3, II, G, the values I x n | > sink, with increasing , below every number s > 0, how- ever small. We state the matter more precisely in the following: 39. Definition. If (x n ) is a given sequence, and if it is related to a definite number in such a way that (* - 6 forms a null sequence \ then we say that the sequence (x n ) converges to , or that it is convergent. The number is called the limiting value or limit of this sequence; the sequence is also said to converge to 1-, and zee say that its terms approach the (limiting) value , tend to , have the limit . This fact is expressed by the symbols x n -> 5 or lim x n = . To make it plainer that the approach to is effected by taking the index n larger and larger, we also frequently write 2 x n ->^ for n -> oo or lim x n . w->o& Including the definition of a null sequence in the new definition, we may also say: x n -> for n -> oo (or lim x n = ) if for every chosen e > 0, we can n >x> always assign a number n Q = n Q (e), so that for every n > , we liave 1 Or (f x n ) or | x n f |; by 10, 5 the result is exactly the same. 2 Read: "x n (tends) towards f for n tending to infinity" in the one case, and "Limit x n for n tending to infinity equals f" in the other. In view of the definitions 40, 2 and 3, it would be more correct to write here "n -> + oo"; but for simplicity the -f sign is usually omitted. 8. Convergent sequences. t>5 Remarks and Examples. 1. Instead of saying "(# n ) is a null sequence", we may now, more shortly, write "x n -> 0". Null sequences are convergent sequences with the special limiting value 0. 2. Substantially, all remarks made in 10 therefore hold here, since we are concerned only with a very obvious generalisation of the concept of a null sequence. 3. By 31, 3 and 38, 5, we have for a > 'Y/a f 1 and -\/n -> 1. 4. If (x n | y n ) -^ (7, then x n -> a and y n -> <r. For both | * a | and also | y n - a \ are ^ \ y n - x n \ , so that both, by 26, ], form null sequences together with (y n x n ). / _ |\n 14365 5. For x n = 1 - - n - , that is, for the sequence 2, ^, y ^ ^ ft , . . . , x n ->> 1, for | x n 1 | forms a null sequence. 6. In geometrical language, x n -> f means that all terms with sufficiently large indices he in the neighbourhood of the fixed point . Or more precisely (cf. 10, 13), m every e -neighbourhood of f, the whole of the terms, with at most a finite number of exceptions, are to be found 3 . In applying the mode of representation of 7, 6, we draw parallels to the axis of abscissae, through the two points (0, f e) and may say : x n - > , if the whole graph of the sequence (x n ), with the exception of a finite initial portion, lies in every s-strip (however narrow). 7. The lax mode of expression: "for n = oo , x n = f" instead of x n -> f, should be most emphatically rejected. For an integer n = oo does not exist and v n need never be f . We are concerned merely with a process of approximation, sufficiently clear from all that precedes, which there is no ground whatever for imagining completed in any form. (In older text books and writings we frequently find, however, the symbolical mode of writing: "lim x n f", to which, since it W~00 is after all meant only symbolically, no objection can be taken, excepting that it is clumsy, and that writing "n oo" must necessarily create some confusion regarding the concept of the infinite in mathematics. 8. If x n -* , then the isolated terms of the sequence (x n ) are also called approximations to , and the difference x n is called the error corresponding to the approximation x n . 9. The name "convergent" appears to have been first used by J. Gregory (Vera circuit et hyperbolae quadratura, Padua 10(37), and "divergent" (40) by Bernoulli (Letter to Leibniz of 7. 4. 1713). It was through the publications of A. L. Cauchy (see p. 72, footnote 18) that a limiting value came to be denoted generally by the prefixed symbol "lim". The arrow sign (->), which is so particularly appropriate, came into common use after 1906, through the works of G. H. Hardy, who himself referred it back to J. G. Leatham (1905). To the definition of convergence we at once append that of diver- gence: Definition 1. Every sequence which is not convergent m the sense 40. of 39 is called divergent. 3 Frequently this is expressed more briefly: In every e-neighbourhood of ? "almost all n terms of the sequence are situated. The expression "almost all" has, however, other meanings, e. g. in the Theory of Sets of Points. 66 Chapter 11. Sequences of real numbers. With this definition, the sequences tt, 2, 4, 7, 8, 11 are certainly divergent. Among divergent sequences, one type is distinguished by its particularly simple and transparent behaviour, e. g. the sequences (n*}> (n), (a n ) for a > 1, (logn), and others. Their common property is evidently that the terms increase with increasing n beyond every bound, however high. For this reason, we may also say that they tend to -| oo, or that they (or their terms) become infinitely large. This we put more precisely in the following Definition 2. // the sequence (# M ) has the property that, given an arbitrary (large) positive number G, another number n Q can always be assigned such that for every n > w then 4 we shall say that (x n ) diverges to |- oo , tends to + oo , or is definitely divergent 5 with the limit + oo ; and we then write x n -> + oo (for n > oo) or lim x n + oo or Km x n + oo. M >' We are merely interchanging right and left by defining further: Definition 3. // the sequence (x n ) has the property that, given an arbitrary negative number G (large in absolute value), another number n Q can always be assigned such that for every n > w *<-<?, then we shall say that (x n ) diverges to oo, tends to oo or is definitely divergent 5 with the limit GO, and we write x n -> oo (for n -> oo) or lim x n oo or lim x n ~ oo. n- Remarks and Examples 1 The sequences (n), (n*), (n n ) for a > 0, (log*), (log n) a for a>0, tend to H-OO; those whose terms have these values with the negative sign tend to OO. 2. In general- If # >-f-oo, then y n f = x n -> QO, and conversely. It is therefore sufficient, substantially, to consider divergence to +CO in what follows. 3. In geometrical language, x n * + OO means, of course, that however a point G (very far to the right) my be chosen, all points x n , except at most a linite number of them, remain beyond it on the right. With the mode of 4 Notice that here not merely the absolute values \x n \, but the numbers x n themselves, are required to be >> G. 6 It is sometimes even said, with apparent distortion of facts, that the sequence converges to -f oo. The reason for this is that the behaviour described in Definition 2 resembles in many respects that of convergence (39). We will not, however, subscribe to this mode of expression, although a mis- understanding would never have to be feared. Similarly for OO. 8. Convergent sequences. 67 representation in 7, 6, it means that* however far above the axis of abscissae we may have drawn the parallel to it, the whole graph of the sequence (x n ) excepting- a finite initial portion, lies still further above it. 4. The divergence to ^ CO need not be monotone; thus for instance the sequence 1, 2 1 , 2, 2 s 3, 2 3 , 4, 2 4 , ..., A, 2*, ... also diverges to + 00. 5. The succession 1, 2, +3, 4, ..., ( l)"~ 1 n, ... does not diverge to -foo or to OO. This leads us to the further Definition 4. A sequence (x n ), which either converges in the sense of definition 39, or diverges definitely in the sense of the defini- tions 40, 2 and 3, will be said to behave definitely (for n+oo). All other sequences, which therefore neither converge, nor diverge defini- tely, will be called indefinitely divergent or, for short, Indefinite*. Remarks and Examples. 1. The sequences [(-I)"], [(-2)"], (a") for a<-l, and likewise the se- quences 0, 1, 0, 2, 0, 3, 0, 4, ... and 0, 1, 0, 2, 0, - 3, . . ., as also the se- quences 6, 4, 8 are obviously indefinitely divergent. 2. On the contrary, the sequence (| n |) for arbitrary a, and, in spite of all irregularities in detail, the sequences (3 n -f-( 2) n ), (n-\- ( l) n log n), (n 9 -j-( l) n w), show definite behaviour. 3. The geometrical interpretation of indefinite behaviour follows imme- diately from the fact that there is neither convergence (v. 39, 6) nor definite divergence (v. 40, 3, rein 3). 4. Both from x n * + 00 and from # n -> oo it follows, provided every term 4= 7 , that -* 0; for | x n \ > G = evidently implies < *. On x n s x n the other hand, x n + in no way involves definite behaviour of (- ) \ x n / (-i) n /n Example: For x n , we have a? n ->0, but ( J indefinitely diver- n \ x n J gent. We have however, as is easily proved, the Theorem: // (x n ) is a null sequent e whose terms all have the same sign, then the sequence ( J is definitely divergent; and of course to -foo or \Xfi/ OO, according as the x n 's are all positive or all negative. 9 We have therefore to consider three typical modes of behaviour of a i equence, namely: a) Convergence to a number f, in accordance with 39; ')) divergence to OO, m accordance with 40, 2 and 3; c) neither of the i wo . Since the behaviour b) shows some analogy with a) and some with c), modes of expressions in use for it vary. Usually, it is true, b) is reckoned as ilivergence (the mode of expression mentioned in the last footnote cannot be consistently maintained) but "limiting values" -J-oo and oo are at the name time spoken of. We therefore speak, in the cases a) and b), of a de- inite, in the case c) of an indefinite, behaviour; in case a), and only in his case, we speak of convergence, in the cases b) and c) of divergence. Instead of "definitely and indefinitely divergent", the words "properly and im- properly divergent" are also used Since, however, as remarked, definite di- vergence still shows many analogies to convergence and a limit is still spoken of in this case, it does not seem advisable to designate this case precisely as that of proper divergence. 7 From some place onwards this is certainly the case (58 Chapter II. Sequences ot real numbers. To facilitate the understanding of certain cases which frequently occur, we finally introduce the following further mode of expression: Definition 5. // two sequences (% n )and(y n } 9 not necessarily con- vergent, are so related to one another that the quotient Xn y n tends, for *-|-oo, to a definite finite limit different from zero 8 , then we shall say that the two sequences are asymptotically proportional and write briefly // in particular this limit is 1, then we say that the two sequences are asymptotically equal and write, more expressively * n ^y- Thus for instance __ I V" a 4- 1 * * * 1 (5 w" + 23) ~ log n , \'n + 1 - ^/n ~v - , 1/n 1 -f 2 H ----- h n ~ ri 2 , l-4-2 2 + .**4-n a ^- j n 3 . These designations are due substantially to P. dw Bois-Reymond (Annali di matematica pura ed appl. (2) IV, p. 338, 1870/71). To these definitions we now attach a series of simple, but quite fundamental Theorems on convergent sequences. 41. Theorem 1. A convergent sequence determines its limit quite uniquely 9 . Proof. If x n +!;, and simultaneously x n ', then (x n f) and (x n ') are null sequences. By 28, 2, is then also a null sequence, i. e. | = f, q. e. d. 10 8 x n and y n must then necessarily be =}= /n?w some place onwards. This is not required for every n in the above definition. 9 A convergent sequence therefore defines (determines, gives . . .) its limit quite as uniquely as any nest of intervals or Dedekind section defines the number to which it corresponds. Thus from this point we may consider a real number as given if we know a sequence converging to it. And as formerly we said for brevity that a nest of intervals (a;,, | y n ) or a Dedekind section (A \ B) or a radix fraction is a real number, so we may now with equal right say that a sequence (x n ) converging to f is the real number f , or symbolically: (# n ) = { . For further details of this conception, which was used by G. Cantor to construct his theory of real numbers, see pp. 79 and 95. 10 The last step in our reasoning, by which the reader may at first sight be taken aback, amounts simply to this: If with respect to a definite numerical value a we know that, for every e>0, we always have | a | < e, then we 8. Convergent sequences. l>y Theorem 2. A convergent sequence (x n ) is invariably bounded. And if | x n | 5g K, then for the limit we have u | | ^ /. Proof. If x n -> , then we can, given e > 0, assign a number m, such that for every n > m f-e<* n < + e. If therefore K l is a number greater than the m values | x l [, | x 2 |, . . . , | x m | , and greater than | | + e, then obviously I * i < *i for every n. Now let K be any bound of the numbers | x n \. If we had | | > K y then | | K > and therefore, from some place onwards in the sequence, \t\-\* n \^\* n -t\<\e\-K and therefore | x n \ > K, which is contrary to the meaning of K. Theorem 2a. x n -> f implies \ x n \ -> \ |. Proof. We have (v. 3, II, 4) therefore ( | x n \ \ \ ) is by 26, 2 a null sequence when (# n ) is. Theorem 3. If a convergent sequence (x n ) has all its terms different from zero > and if its limit g is also 4= 0, then the sequence () is bounded; \x n / or in other words, a number y > exists, such that \ x n \ ^ y > for every n; the numbers \ x n \ possess a positive lower bound. Proof. By hypothesis, J | | = e > 0, and there exists an integer m, such that for every n > m, \ x n | < e and therefore | x n \ > % \ g \ 12 . If the smallest of the (m + 1) positive numbers | x |, | x 2 |, . . . , | x m \ and i | ^ | be denoted by y, then y > 0, and for every n, \ x n \ ^ y, = l -, q. e. d. If, given a sequence (# n ) converging to , we apply to the null se- quence (x n ) the theorems 27, 1 to 5, then we immediately obtain the theorems: necessarily have a 0. For is the only number whose absolute value is less than every positive e. (In fact | | < c is true for every e > 0. But if a 4= 0, so that | a | > 0, then | a | is certainly not less than the positive number e = J | a |.) Simi- larly, if we know of a definite numerical value a that, for every e > 0, we always have a ^ K + e, then we must have further a g K. The method of reasoning involved here: "If for every z > 0, we always have | a | < e, then necessarily a 0" is precisely the same as was constantly applied by the Greek mathematicians (cf. Euclid, Elements X) and later called the method of exhaustion 11 Here the sign of equality in "| f | *g K" must not be omitted, even when, for every n t \ x n \ < K. 12 For n ^ m, all the x n 's are therefore necessarily 4= 0. 70 Chapter II. Sequences of real numbers. Theorem 4. // (x n ') is a sub-sequence of (x n ), then x n +t; implies x n '>. Theorem 5. // the sequence (x n ] can be divided into two sub- sequences of which each converges to , then (x^ itself converges to . Theorem 6. // (#') is an arbitrary rearrangement of x n , then x n -> implies x n ' -> f . Theorem 7. // x n >f and (# n ') results from (# n ) by a finite number of alterations, then x n '+$. Theorem 8. // # n ' *l and # n "-*f, and if the sequence (x^ is so related to the sequences (x n ') and (x n ") that from some place onwards, (i. e. for every n^>m, say t ) rr ' < r < v " X n ^ X n ^ X n > then x n +. Calculations with convergent sequences are based on the following four theorems: Theorem 9. x n > f and y n * r\ always implies (x n + y n ) ~ * f + ^ * and the corresponding statement holds for term by term addition of any fixed number say p of convergent sequences. Proof. If (x n |) and (y n rf) are null sequences, then so, by 28, 1, is ((x n + y n ) (f -{- 17)). In the same way, 28, 2 gives the Theorem 9 a. x n + f and y n +r], always implies (x n y n ) + f rj . Theorem 10. x n >| and y n +ri> always implies x n y n +r)> and the corresponding statement holds for term by term multiplication of any fixed number say p of convergent sequences. In particular: x n * implies cx n +c, whatever number p denote. Proof. We have and since here on the right hand side two null sequences are multi- plied term by term by bounded factors and then added, the whole expression is itself the term of a null sequence, q. e. d. Theorem 11. x n * and y n +r] always implies, if every x n =^Q and also f 4= yn jv **~*f ' Proof. We have yn ri ___ y S-x H rj __ (y n ~ >y) g - (x n - g) 17 18 Or three, or any definite number. 8. Convergent sequences. 71 Here the numerator, for the same reasons as above, represents a null sequence, and the factors - are, by theorem 3, bounded. Therefore S' x n the whole expression is again the term of a null sequence. Only a particular case of this is the Theorem 1 1 a 14 . x n - > g always implies, if every x n and also are 4=0, , ! These fundamental theorems 8 11 lead, by repealed application, to the following more comprehensive Theorem 12. Let R = R (x (l) , z 2) , x (3) , . . ., xW) denote an ex- pression built up, by a finite number of additions, subtractions, multi- plications, and divisions, from the letters x (1) , o? (9) , ... 9 x ( &, and arbitrary numerical coefficients 1 *; and let be p given sequences, converging respectively to (1) , f (3) , . . ., f & } . Then the sequence of the numbers provided neither in the evaluation of the terms R n , nor in that of the number R( (1) , (2) , > (p) ), division by is anywhere required. These theorems give us all lhat is required for the formal mani- pulation of convergent sequences: We give a few more Examples. 1. -> implies, if a>0, invariably, 42, a*"-*a. For a*- a* ^ (<***-*-- x ) fs a null sequence by 35, 3 2. a:-*-f implies, if every . and also are >-0, that log ac m -* log | . Proof. We have log o:,, - log log ^ = loff (l + *" which by 38, 7 is a null sequence, since x n > implies ~-~- > 1 . 14 In theorems 3, 11 and lla, it is sufficient to postulate that the limit of the denominators is 4= 0, for then the denominators are, from some index m on- wards, necessarily 4= 0, and only "a finite number of alterations" need be made, or the new sequence need only be considered for n > m, to ensure this being the case for all. lfi More shortly: a rational function of the /> variables .\; (1) , v w , . . . , .x- (p) with arbitrary numerical coefficients. 72 Chapter II. Sequences of real numbers. 3. Under the same hypotheses as in 2., we also have, for arbitrary real Q, Proof. We have \ x% Q = l \ which by 38, 8 is a null sequence 16 , since *"-p-^ > 1 and tends to as n ~> QO. (This is to a certain extent further completed by 35, 4.) Cauchy's theorem of limits and its generalisations. There is a group of theorems on limits 17 essentially more pro- found than the above, and of great significance for later work, which originated in their simplest form with Cauchy 16 and have in recent times been extended in different directions We have first the simple 43. Theorem 1. // (# , x 19 ...) is a null sequence, then the arith- metic means ~. 9 %o ~r x i ~r "r % r\ - Q X n n-f- 1 ' * * *' also form a null sequence. Proof. If s is given > 0, then m can be so chosen, that for every n > m we have la; I < -- . For these n's, we then have n - n+1 T- 2 n + 1' Since the numerator of the first fraction on the right hand side now contains a fixed number, we can further determine n , so that for n > w that fraction remains < -|-. But then, for every n > n Q , we have | # n ' | <C e, and our theorem is proved. Somewhat more general, but nevertheless an immediate corollary of this, is the Theorem 2. // x n *, then so do the arithmetic means ltt Examples 1. to 3. mean in the language of the theory of functions that the function a x is continuous at every point, the functions log a; and X Q at every positive point. 17 The reader may defer the study of these theorems until, in the later chapters, they come into use. 18 Augustin Louis Cauchy, born 1789 in Paris, died 1857 in Sceaux. In his work Analyse alg&bnque, Paris 1821 (German edition, Berlin 1885, Julius Springer) the foundations of higher analysis are for the first time developed with full rigour, and among them the theory of infinite series. In what follows we shall frequently have to refer to it; the above theorem 2 may be found on p. 59 of that treatise. 8. Convergent sequences. 73 Proof. By theorem 1, /fa-e + fe-g)+...+(*.-m = ^ ' _ |) is a null sequence when (x n ) is, q. e d. From this theorem, the corresponding one for geometric means now follows quite easily. Theorem 3. Let the sequence (y , y a , . . .) ^77, and have all its members and its limit r\ positive. Then also the sequence of geo- metric means / -,"/: Proof. From y n ^, since all the numbers are positive, we deduce, by 42, 2, that By theorem 2, it follows that a; _ *i*+ n -+* _ log ^ yi y 2 ...^ = log y n '- log 17. By 42, 1, this at once proves the truth of our statement. Examples. '+!'- -4 1. ----- *0, because -- *0. w n 2. V f = l--"-~-* 1 ' because n_ yn n _ 8. - - - - - - -- > 1, because y n -* 1 . / i \ 4. Because ( 1 -^ -- ) - (v. 46 a in the next ), we have by theorem 3, . -2 - V^T- - --" also or, therefore, i n _ i JLy M !^, n r e ' n _ n a relation which may also be noted in the form "^n\^. ". 74 Chapter II. Sequences of real numbers. Essentially more far-reaching, and yet as easily proved, is the following generalisation of Cauchy's theorems 1 and 2, due to $. Toeplitz: Theorem 4. Let (X Q , #j , . . .) be a null sequence and suppose the coefficients a flv of the system (A) satisfy the two conditions: (a) Every column contains a null sequence, i. e. for fixed P^O fl np >0 when n >-{-oo. (b) There exists a constant K, such that the sum of the absolute values of the terms in any one row , i. e., for every n, the sum kaol + KiH ----- \-\a nn \ remains < K. Then the sequence formed by the numbers X n = <*nO X + *nl X l + a n* X * H ----- H <*nn X n is also a null sequence. Proof. If e is given > 0, determine m 50 that for every n>m \ x \ < ^- 'Ihen for those w's, By the hypothesis (a), we may now (as m is fixed) choose n > m, so that for every n > n {} , we have | a nQ x ~{ ----- 1- a nm x m \ < y . Since for these w's | x n ' \ is then -< e, our theorem is proved. In applications it is useful to have the following Complement. If, for the coefficients a^ t are substituted other numbers a^ = a*i *^ x ^, obtained from the numbers a^\ by multiplication 19 Cauchy's Theorem 1 has been generalised in several ways, in particular by J. L. W. V. Jensen (Cm en Satning af Cauchy, Tidsknft for Mathematik, (5) Vol. 2, pp. 81 84. 1884) and O. Stolz (t)ber erne Verallgemeinerung eines Satzes von Cauchy, Mathemat. Annalen, Vol. 33, p. 237. 1889). The above formulation, due to O. Toephtz (Uber hneare Mittelbildungen, Prace matematycznofizyczne, Vol. 22, p. 113 119. 1911), is in a certain sense a final generalisation, for this reason that it shows (1. c.) the conditions, recognised in Theorem 5 as sufficient, to be also tiecf \\ary, tor \ n - - to imply x n ' -* in all cases (cf. 221, and the work of /. Sthur: Cber hneare Transformationen in der Theorie der unendlichen Reihen, Jour. f.d. reine u. angew. Math., Vol. 151, pp. 79111. 1920). 8. Convergent sequences. 75 by factors X A M in absolute value less than a fixed constant a, then the numbers /0m a null sequence. Proof The a^'s also satisfy the conditions (a) and (b) of theorem 4; for, if p is fixed, a' np -+Q by 26, 1, and the sums o + i+"- + n remain <K=--aK. From Theorem 4 we may now deduce the Theorem 5. // x n >f, a^ tfci coefficients a^ v satisfy, besides the conditions (a) an^ (b) of Theorem 4, //t further condition also the sequence formed by the numbers Proof. We now have whence our statement at once follows, in consequence of condition (c), by theorem 4. Before giving examples and applications of these important theorems, we may prove the following further generalisation, which points in a new direction. Theorem 6. // the coefficients a fiv of the system (A) satisfy, besides the conditions (a), (b) and (c) mentioned in Theorems 4 and 5, the further condition, that (d) the numbers in each of the "diagonals" of A form a null sequence, i.e. for fixed p, a nn _ p >0 when n+-\-<x>, then it follows from x n -^> and y n +r] that the numbers Proof. Since x v = (x - "V jnv \~v we have n Z n = J|j #n' ynv\Z v=0 10 In the applications, we shall generally have A n = 1. 11 For positive a^v, this theorem may be found in a paper by the author "Uber Summen der Form a b n -f a t 6,,^ + - -f- a w 6 " (Rend, del circolo mat. di Palermo, Vol. 32, p. 95-110. 1911). 76 Chapter II. Sequences of real numbers. Here the first sum tends to zero, by Theorem 4 and its complement, for (x v f) is a null sequence and the factors y n _ y are bounded. And if the second sum be written in the form v=0 v=0 we see, by theorem 5, that this, and thereby also z n , tends +$rj; for the numbers a' nv = a nn - v satisfy, in consequence of (d), precisely the condition (a) there stipulated. 44. Remarks, applications and examples. 1. Theorem 1 is a particular case of Theorem 4; we need only put, in the latter, a0 = a ni = - = a nn = ^~T\ > (n = 0, 1, 2, . . ) Theorem 2 is derived in the same way from Theorem 5. The conditions (a), (b), (c) are fulfilled. 2. If , a a , . . . are any positive numbers, for which the sums it follows 22 from x n -> f that In fact, we need only put, in theorem 5, a _x / = 0, 1,2, ... n '~o n \ - = 0,1 n to see that the statement is correct. The conditions (a), (b), (c) are fulfilled. For <x n == 1 , we again obtain Theorem 2. 2 a. The theorem of no. 2. remains true for f = + oo or f = oo . The same remark holds for the general theorem 5, provided all the a^v's are >: there. For if x n + + oo and, as in the proof of Theorem 4, m be so chosen, given G>0, that for every n^>m we have rt > G-|- 1, then for those n's we have x n ' >(G -f 1) (a mm+l + . . . -f a nn ) - a no \ x \ - . . . - a n m \ x m \ . In consequence of the conditions (a) and (c) in Theorems 4 and 5, we may therefore so choose n that for every n>w we have aj/>G. Hence *'-> + 00. u 3. Instead of assuming the a n 's positive and a n -> + GO , it suffices [by (b)] to require onlv that | a | -f | a, | + . . . + | a n | -> -f O, with the proviso, however, that a constant K exists, such that 23 for every n I o I + I i I + . - + I n I < K ' | a + a, + . . . + a n |. (For positive a n , J=l gives all that is here required.) 84 O. 5<o/z, loc. cit. Of course it also suffices, that the a n 's be from some point onwards >0, provided only o n ->-foo. The # n "s must then be con- sidered from that point onwards, after which a n is > 0. 88 Jensen, loc. cit. If oc m is the first of the a's to be =j= 0, then the x m "s are defined only for w>w. 8. Convergent sequences. 77 4. If in 2. or 3 we put, for brevity, ce n x n y ny then we obtain: o i - and provided the a n 's satisfy the conditions given in 2. or 3. 5. If we write further y +y l -f- . - . +y n = ^n, and + <* t + + ** = 4 then the last result takes the form: v v _ v -*, provided J-_Jua-*f, -^n ^n -n i and provided the numbers cc n = A n A n _^ (n > 1, a = ^4 ) satisfy the conditions given in 2. or 3. 6. Thus we have, for instance, by 5.: .. 1 + 2-I-...4-* .. n n 1 lim - K - = lim - = hm s - - = -^ w^ w a (n l) a 2n 1 2 Similarly we have l + 2 a +...+n 9 ,. n* 1 lim ---- - --- = lim -- - - -- - = n 3 w a -(w I) 3 3 and generally -^-. 2 ,. L F -f- Z r -j- . . . -J- W lim : ! = lim n* * lim- it p denotes a positive integer. 7. Similarly we find, if we anticipate the proof in 46 a of the convergence of the sequence of numbers ( 1 -\ ) ! \ n / log 1-f- log 2 -{-...+ log n lognl . V . ! ^ = . 1 I.e. Ino- -M! /-v/ loort _ ^ nlogn log n n 8. The numbers fulfil the conditions (a), (b) and (c) of the theorems 4 and 5; for if p be fixed, *np+Qi seeing that it is = ' and therefore < l ( v - 38 ' 2 > while toi every n. Therefore o; n --? always implies 78 Chapter II. Sequences of real numbers. 9. The same specialisations as were given in 1., 2., 3. and 8. for theorem 5 may of course also be applied to theorem 6. We merely mention the two following theorems: (a) From x n *> and y n -^-rj it always follows that x, ?__! 4- a-g y n _ 2 -f ... j-x n y fr (b) If (#) and (y n ) are two null sequences, the second of which fulfils the extra condition that for every n remains less than a fixed number K, then the numbers form a null sequence. (For the proof we put a nv = y n - v in theorem 4.) 10. The reader will have noticed that it is in no wise essential that the rows of the system (A) of theorem 4 should break off exactly at the n ih term. On the contrary, these lows may contain any number of terms. Indeed, after we have mastered the first principles of the theory of infinite series, we shall see that these rows may contain even an infinity of terms (a no , /tlT . . ., a nv , . . .), provided only the other conditions imposed on the system be fulfilled. The theorem hereby indicated will be formulated and proved in 221. 9. The two main criteria. We are now sufficiently prepared to attack the actual problems of convergence. There are two mam points of view from which we propose, in what follows, to examine the sequences which come before us. We have above all to consider the Problem A. 7s a given sequence (x n ) convergent, or definitely or indefinitely divergent? (Briefly: How does the sequence behave with respect to convergence?) And if a sequence has pioved to be convergent, so that the existence of a limiting value is ensured, we have further to consider the Problem B. To what limit $ does the sequence (# n ), recognized to be convergent, tend? A few examples may make the significance of these problems clearer: If for instance we are given the sequences examination of their construction shows that there are always two (01 more) forces which here, so to speak, oppose one another and thereby call forth the variation of the terms. One force tends to increase, 9. The two main criteria. 79 the other to diminish them, and it is not clear at a glance which of the two will get the upper hand or in what degree this will happen. Every means which enables us to decide the question of convergence or divergence of a given sequence, we call a criterion of convergence or of divergence; these serve, therefore, to solve the problem A. The problem B is in general much more difficult. In fact, we might almost say that it is insoluble, or else is trivial. The latter, because a convergent sequence (x n \ by theorem 41, 1, entirely deter- mines its limit f , which may therefore be regarded as "given" by the sequence itself (cf. footnote to 41, 1). On account, however, of the boundless complexity and multiplicity of form which sequences show, this conclusion does not seem very satisfactory. We shall wish, rather, not to consider the limit | as "known", until we have before us a Dedekind section, or still better a nest of intervals, for instance a radix fraction, in particular a decimal fraction. These latter especially are the methods of representing a real number with which we have always been most familiar. If we regard the problem in this light, we may call it the question of numerical calculation of the limit 1 . This question, one of great practical significance, is usually in theoretical considerations of very second-rate imporiance, for from a theoretical point of view, all modes of representation for a real number (nests, sections, sequences, . . .) are precisely equivalent. If we observe further, that the representation of a real number by a sequence may be considered as the most general mode of representation, our problem B may be stated in the following form. Problem B'. Two convergent sequences (#J and (#) are given, how may we determine whether or not both define the same limit, or whether or not the two limits stand in a simple relation to one another? A few examples will serve to illustrate the kind of question referred to: l - Let _/i J.1Y 1 ^ ' fi^ *\ 45. Both sequences are quite easily (v. 46 a and 111) seen to be convergent. But it is not so apparent that if denotes the limit of the first sequence, that of the second is = f *. 2. Given the sequence _!_ 3 ^ 17 41^ 1 ' ~2~' 5 ' "12 ' 29 ' " in which the numerator of each fraction is formed by adding twice the nume- rator of the last fraction preceding to the numerator of the last fraction but one (e. g. 41 = 2-17 + 7), nnd similarly for the denominators. The question of 1 Numerical calculation of a real number = representation of that num- ber by a decimal fraction. For further details, see chapter VII T. 80 Chapter II. Sequences of real numbers. convergence again gives no trouble, nor does the numerical evaluation of the limit, but how are we to recognise that this limit and let x n be the perimeter of the regular polygon with n sides inscribed in the circle of radius 1. Here also both sequences are easily seen to be con- vergent. If f and f are their limits, how does one see that here ' = 8? These examples make it seem sufficiently probable, that Problem B or B' is considerably harder to attack than Problem A. We therefore confine our attention in the first instance entirely to the latter, and to begin with make ourselves acquainted with two criteria, from which all others may be deduced. First main criterion (for monotone sequences). 46. A monotone bounded sequence is invariably convergent; a mono- tone sequence which is not bounded is always definitely divergent. (Or, therefore: A monotone sequence always behaves definitely, and is then and only then convergent, when it is bounded, and then and only then divergent, when it is not bounded. In the latter case the diver- gence is towards -f- oo or oo according as the monotone sequence is ascending or descending.) Proof, a) Let the sequence (# n ) be monotone ascending and not bounded. Since it is then (because x n ^ o^) certainly bounded on the left, it cannot be bounded on the right; given any arbitrary (large) positive number G, there is then always an index n Q , for which But then, since the -sequence is monotone increasing, we have for every n > n , a fortiori, x n > G, and so, by Definition 40, 2, actually x n -+ + oo. Interchanging right and left, we see in the same way that a monotone descending sequence which is not bounded must diverge to oo. Thus the second part of the proposition is also proved. b) Now let (x n ) be a monotone ascending, but bounded sequence. There is then a number K, such that | x n \ <; K for every n, so that for every n. The interval J = aj x . . . K therefore contains all the terms of (x n ); to this interval we apply the method of successive bisection: We denote the right or the left half of / by / 9 , according as the right half does or does not still contain points of (#J. From / we select one half by the same rule, and call this / 3 ; and so on. The intervals of the nest so constructed have the properly 2 , that no point The reader should illustrate the circumstances on the number-axis. 9. The two mam criteria. 81 of the sequence lies to the right of them, but at least one lies inside each of them. Or in other words: the points of the sequence (while monotonely progressing towards the right) penetrate into each interval, but do not emerge from it again; in each of these intervals, therefore, all points from a certain index onwards come to lie. We may there- fore, if we suppose the numbers n if n^ y . . . properly chosen, say that: In J k lie all x n 's with n > n k , but to the right of J k lie no more x n 's. If f is now the number determined by the nest (/ M ), it can at once be shewn that a; w *f. For if e is given > 0, choose the index p so that the length of / is less than 6. For n > n p , all the x n 's lie, together with f , in / , so that for these ns we must have (x n ) is therefore a null sequence, and x n * , q. e. d. By a suitable interchange of right and left, we see that monotone descending bounded sequences must also be convergent. Thus every part of the theorem is proved. Remarks and Examples. 1. We first draw attention again to the fact that (cf. 41, 1) even when I %n | <Z K , we may have for the limiting- value the equality | | = K . 2. Let As the sequence is monotone increasing-, and as x n <n> - T^^ ^ * s a l so bound- n -f- 1 ed. // is therefore convergent. Of its limit f we know no more, so far, than that 37 for every n, which e. gf. for n = 3 becomes ~^< 1. Whether it has a ra- tional value, or whether bears a close relation to a number appearing in any other connection in short: an answer to problem B cannot here be perceived at once. Later on we shall see that is equal to tt e natural logarithm of 2. I. e. the logarithm of 2 whose base is the number e introduced in 46a below. 3. Let o? n =(l+- s - + -5--f-H ), so that the sequence (x n ) is monotone \ Jo n/ increasing (cf. 6, 12). Is it bounded or not? If G is given arbitrarily > 0, chose w>2; then for n>2 OT The sequence is therefore not bounded and consequently diverges + -f oo . 82 Chapter II. Sequences of real numbers. 4. If o = (x n | y n ) is an arbitrary n'est of intervals, the left and right end- points of the intervals respectively form two monotone, bounded and therefore convergent sequences. We then have lim r n = hm y n = (x n \ y n ) = a . 16a. As a particularly important example, we will consider the two sequences whose terms are __- .*\ ^ v ~\ J^AAI/ * ; *i x ^v ~' \ ' " and = * *.- We have no means of perceiving immediately (cf. the general remark on p. 78) how the sequences behave as n increases. We proceed to show first that the second sequence is monotone descending, that is to say that for n I> 2 This inequality is in fact equivalent 3 to or to But the truth of //w's inequality is evident, since, by Bernoulli's in- equality 10, 7 we have, for a > I, a --\= and every n > 1, or in particular As, moreover, y n > 1 for every n, the sequence (yj is monotone des- cending and bounded, and therefore convergent Its limit will ofter occur later on; it is, since Rulers time, denoted by the special 4 letter e. As regards this number, we can only deduce for the present that which for e. g. n = 5 becomes . ^ 6 6 3 That is to say, each inequality follows from all the others. 4 Euler uses this letter to designate the above limit in a letter to Goldbach (2.*). Nov. 1731) and in 1736 m his work: Mechanica sive motus scientia analytice exposi ta, II, p. 251. 9. The two main criteria. 83 The first of our two sequences, on the contrary, is monotone ascending. In fact, # n _x < x n here means 5 (1 \n--l / l\n 1 + ,T--] or ' e- l~^< (" 2 7 T = ( l ~ Vf- n \ n 2 / \ v But, again by 10, 7, we have actually for every # > 1, n+i The sequence (x n ) is therefore monotone increasing. As, in any case, (1 \ n i+ i) we have, for every w, A: W ^JVi, i- c. (jc w ) is also bounded and hence con- vergent. As, finally, the numbers are all positive and (by 26, 1) form a null sequence, we conclude at once that (x n ) has the same limit as (j' n ). Thus lim x n = lim j> n = . And for this number e we have furthermore, as has appeared in the proof, in a nest of intervals defining it. (It provides, for instance taking n 3, the inequality of < e < 2 H 5 t -; we shall however become acquainted later on ( 23) with other sequences converging to e y which are more convenient for numerical calculation.) This is the number e that (cf. p. 58) forms the base of the natural logarithms. We shall accordingly agree to use the symbol log to mean this natural logarithm to the base e, unless the contrary is expressly stated. The fruitfulness of the first main criterion is due above all to the fact that it allows us to deduce the convergence of a sequence of numbers from very few hypotheses, and these such as are usually very easy to verify namely, from monotony and boundedness alone. On the other hand, however, it still relates only to a special, even though particularly frequent and important kind of sequence, and therefore 5 Cf. footnote 3. 84 Chapter IT. Sequences ot real numbers. appears theoretically insufficient. We shall therefore ask for a criterion which enables us to decide quite generally as to the convergence or divergence of any sequence. This is accomplished by the following 47. Second main criterion (1 st form). An arbitrary sequence (x n ) is convergent if and only if, given s > , a number n = n (e) can always be assigned , such that for any two indices n and n' both greater than n , wz have in every case \X n X n >\<e. We first give a few Explanations and Examples. 1. The remarks 10, 1, 3, 4 and 9 are also substantially applicable here; and the reader is recommended to read them through once more in this con- nection. 2. The criterion states to put it in intuitive language: all o; n 's with very high indices must he very close together. 3. Let o; = 0, x t = 1 , and let every term after these be the arithmetic mean between the two terms which precede it, i. e. for n"^2 s so that x a = , # 8 = , o; 4 = jj r , . . .. In this evidently not monotone sequence it is clear, on the one hand, that the differences between consecutive terms form a null sequence; for it may be verified quite easily by induction that _ x n + 1 x n 2" and so tends to 0. On the other hand, between these two consecutive numbers all the following ones lie. If therefore, after s has been assigned ^>0, we choose p so large that < , we have 2 p I X n - X n ' |< 9 provided only n and n' are ^>p. By the 2 nd mam criterion the sequence (x n ) is therefore convergent. The limit { also happens to be easily obtainable. A little reflection in fact leads to the surmise that f = J-. In point of fact, the formula 2 ~ 2 (n 1 )** 1 * w ~3~3* 2* can immediately be proved by induction and shows that x n is actually a null sequence. Before trying to fathom the meaning of the 2 nd main criterion further, we proceed to give its Proof, a) That the condition of the theorem let us call it for brevity its e condition is necessary, i. e. that it is always fulfilled 8 This is true for n = and 1. From g fc+a -s fc + 1 = ** + * ^-f.S-JIJLzA & & it follows that if proved for every n < k , it is true for n = k -|- 1 . 9. The two mam criteria. 85 if (# M ) is convergent, is seen thus: If ", then (x n - ) is a null sequence; given e > 0, we can so choose n that for every n > n , | x n | j is < -- . If besides w, we also have n > W , then | ;r n j is also < -^ , and so which proves this part of the theorem. b) That the e- condition is also sufficient is not so easy to see We again prove it constructively, by deducing from the sequence (x n ) a nest of intervals (/J and then showing that the number determined thereby is the limit of the sequence. This is done as follows: Any e > being chosen, | x n x n ' \ must always be < e provided only the indices n and n' both exceed some sufficiently large value. If we suppose the one fixed and denote it by p, then we may also say: Given any e > 0, we can always assign an index p (actually, as far to the right as we please) so that for every n > P If we choose successively = -77, -r , ..., 7^, ... then we get: & 4 u 1) There is an index p^ such that for every n > p 1 , we have | x n x pi \ < ~ . 2) There is an index p t , which we may assume > p lt such that for every n > p^ , we have | x n x pi \ < -^ , and so on. A th step of this kind gives: k") There is an index p k , which we may assume > p k ^ If such that for every n > p k , we have | x n x p | < o* Accordingly we form the intervals J k : 1. The interval x Pl | . . . x Pi -f- | call / t ; it contains all the x n 's for n>p lt in particular, therefore, the point x p ^. It therefore contains in whole or part the interval x Pt . . . # P2 + J, in which all x n 's with n >> p. 2 lie. As these points also lie in / x , they lie in the common part of the two intervals. This common part we denote 2) by / a and may state: / 2 lies in f and contains all points x with n > p^. If in this result we replace p^ and p^ by p k _ and p k> and denote therefore k) by / fc the portion of the interval o:^ ^ . . . x p ^ + ^ which lies in / k-1 , we may then state: / fc lies in / fc-1 and contains all points x n with n > fc . 4 (G51) 86 Chapter II. Sequences of real numbers. But (J k ) is then a nest of intervals; for each interval lies in the k> preceding and the length of J k is <^ ~ . Now if f is the number thus determined, we assert, finally, that *-* In fact, if an arbitrary e > be now given, we choose an index r so large that < e. We then have & for every n > p r , | x n f | is " < e , since f, together with all x n 's for n > p r , lies in / r and the length of J r is < e. This proves all that was required 7 . Further examples and remarks. 48. 1. The sequence 45, 3 can easily now be seen to be convergent For we have here, if n'>w: I 1 ( l)'-n-i\ If inside the bracket, we take the successive terms in pairs, we see (cf. later 81 c, 3) that the value of the bracket is positive, so that It we now let the first term stand by itself and t ike the following- terms in pairs, we see further that Therefore | ay x n \ is <, provided n and n f are both > ;j The sequence u is therefore convergent. 2. If # = (l -f- -- + H J , we have already seen in 46, 3 that (x n ) is not convergent. With the aid of the 2 nd main criterion, this is dcducible fiom the fact that here the e- condition is not satisfied for << . For however n It may be chosen, we have for n > w and n' 2 n (also therefore > w ) not therefore < 8. The sequence is therefore divergent, and in fact definitely divergent, since it is evidently monotone ascending. 3. The previous example shows at the same time that the contrary of the fulfilment of the $- condition is the following (cf. also 1O, 12)": Not for every choice of s>0 can n be so assigned that the e- condition is then fulfilled; there exists on the contrary (at least) one particular number e > such that, 7 We shall become acquainted with other proofs of this fundamental cri- terion. The proof given above leads immediately to the definition of the limit by the aid of a nest of intervals. A critical account of earlier proofs of the criterion may be found in A. Pnngsheim (Sitzungsber. d. Akad. Mlinchen, Vol. 27, p. 303. 1897). 9. The two mam criteria. 87 above every number n , however large (therefore infinitely often) two positive in- tegers n and n f may be found for which 4. The 2 nd main criterion is now usually, after P. du Bois Reymond (Allge- meinc Funktionentheorie, Tubingen 1882), called the general principle of conver- gence. In substance, it originated with B. Bolzano (1817, cf. O. Stolz, Mathem. Ann. Vol. 18, p. 259, 1881) but was first made a starting point, as an expressly formulated principle, by A. L. Cauchy (Analyse algebrique, p. 125). Our main criterion may also be given somewhat different forms, which are sometimes more convenient in applications. We suppose the notation for the numbers n and n f so chosen that n > n, and therefore we may write n' = n + k , where k is again a positive integer. We then formulate thus the Second main criterion (Form la). 49. The necessary and sufficient condition for the convergence of the sequence (x n ] is that, given any e > 0, a number n = n (e) can always be assigned so that for every n > n and every k^>l we always have From this statement of the criterion we can draw further con- clusions. If we suppose quite arbitrary natural numbers k lf & 2 , . . . , k n , . . . chosen, then we must have, in view of the above, for every n > n I *+*- *!< But this implies that the sequence of differences forms a null sequence. In order to make ourselves more readily understood, we will call the sequence (d n ) for short a difference-sequence of (X M ). In it, d n is therefore the difference between x n and some de- finite later term. Our criterion may then be formulated thus: Second main criterion (2 nd form). 59. The sequence (#J is convergent if and only if every one of its difference-sequences is a null sequence. Proof. The necessity of this condition we have just proved; we have still to show that it is sufficient. We accordingly assume that every difference-sequence tends to 0, and have to show that (x n ) con- verges. But if (# n ) were divergent, there would, by 48, 3, exist a par- ticular number e such that above every number n Q} however large, two numbers n and n' = n -f- k would always he, for which the difference * was 88 Chapter II. Sequences of real numbers. Since this must be the case infinitely often, "there would in contradic- tion to the hypothesis exist difference-sequences 8 which did not tend to 0; (x n ) must therefore converge, q. e. d. Remark. If (,v w ) is convergent, and we choose a particular difference-sequence (d n ), we therefore certainly have d n -> 0. But it should be expressly emphasized that from d n -> alone the convergence of (x n ) need not follow. On the contrary, for this, it is only sufficient that every arbitrary difference-sequence (not merely a particular one) should prove to be a null sequence. If for instance the sequence (1, 0, 1,0, ],...) is considered, every difference- sequence for which all k n y s (from some point onwards) are even numbers is a null sequence. Nevertheless the sequence in question is not convergent. Similarly in the divergent sequence (x n ) with x n 1 + J -f- . . . -f- evety difference-sequence for which the indices k n are bounded forms a null sequence. Extending somewhat further the last obtained formulation of the criterion, we may finally formulate it thus: 51. Second main criterion (3 rd form). If "i> i'2 - > y> is any sequence of positive integers 9 which diverges to -|- GO, and k^ k 2 , . . . , k ni . . . are any positive integers (with- out any restriction), and if we again call the sequence of differences n ~ X 'n+ l *n X 'n for short a difference-sequence of (x n ), then for the convergence of (x n ) it is again necessary and sufficient that (d n ) is in every case a null sequence. Proof. That this condition is sufficient is obvious from the pre- ceding form of the criterion, since (d ri ) must, in the present case also, always be a null sequence when v n is chosen n. And that it is necessary may at once be seen. For if e is chosen > 0, there certainly exists, if (x n ) is convergent (v. Form la), a number m, such that for every n > m and every k ^ 1, we have I v Y I ^ I ^n+k ^n I ^- As v n diverges -> -|- GO, there must be a number n such that for n > H O , we have always v n > m. But then, by the preceding, we have, for n > w , always i. e. (d n ) is a null sequence, q. e. d. 8 For if we denote by n it n 2 , j, . . . the infinite number of values of n for which that inequality (each time with a suitable choice of k) is assumed to be poss- ible, a difference-sequence would exist whose Wj th , w a th , 3 lh , . . . terms were all in absolute value ^ z () ^ 0. This could not then be a null sequence. & Equal or unequal, monotone or not monotone. 10. Limiting- points and upper and lower limits. 89 10. Limiting points and upper and lower limits. The concept of the convergence of a sequence of numbers as defined in the two preceding paragraphs admits of another, some- what more general mode of treatment, by which we shall at the same time become acquainted with some other concepts, of the utmost importance for all that comes after. In #9, 6, we have already illustrated the fact of a given sequence (# ) being convergent by saying that every e- neighbourhood (however small) of f must contain all the terms of the sequence with the possible exception of a finite number at most. There is therefore in eveiy neighbourhood of , however small, certainly an infinite number of terms of the sequence. For this reason, f may be called a limiting point or point of accumulation of the given sequence. Such points may, as we shall at once see, occur also in the case of divergent sequences, and we define therefore quite generally: Definition. A number shall be called a limiting point* of a given sequence (#J if every neighbourhood of f , however small, contains an infinite number of the terms of the sequence; or, therefore, if, for any chosen e > 0, there is always an infinite number of indices n for which Remarks and examples. 1. The distinction between this defimt.on and the definition of limit given 53. in 5JO lies, as already indicated, in the fact that here | r n | <^F needs to be ful- filled not for every n after a certain point, but only for any infinite number of w's, and therefore in particular for at least one n beyond every w . On the other hand, in aceoi dance with &9, the limit of a conveigent sequence (.) is always a limiting point of the sequence. 2. The sequence 6, 1 has the limiting; point 0; 6, 4, the limiting points and 1. (Every number which occurs an infinite number of times in a sequence (x n ) is ipio iacto a limiting point.) 6, 2, 7 and 11 have no limiting point; 6, 9 and 10 have the limiting point 1. 3. We now form an example of more than illustrative significance: If p is an integer > 2, there is obviously only a finite number of positive fractions for which the sum of numerator and denominator = />, namely the fractions - I , .... . Of these we suppose left out all those which are not 1 & p 1 in their lowest terms, and now consider in succession all the fractions thus formed for p = 2, 3, 4, . . . . This gives the sequence, beginning with W 1,2, 1,3, -1,4, 2>-J.T ..... which contains all positive rational numbers. If after each of these numbers we insert the same number with sign changed and start with as first term, we have in the sequence * German: Jliiiifunffswert, Haitfungspunkt or Hdu/ungsstelle. (Tr.) 90 Chapter II. Sequences ot real numbers. (b) 0, 1, -1, 2, -2, *-, -A, 3, -3, J-, -3.4, -4, A _A JL _JL JL 2 ' "2*3' 3 * 4 ' ' ' ' thus formed obviously all rational numbers occurring, each exactly once. For this remarkable sequence every real number is a limiting point; for every neighbourhood of every real number contains an infinity of rational numbers (cf. p. 12). 4. We shall frequently make use of the principle of arrangement in order applied in this example We therefore formulate it somewhat more generally: Suppose that for every k of the series k = 0, 1, 2, ... a sequence <fc) (fc) (&) /Jfe 1 P 1 Jtfn > &< ) && t * * * (.** ~~ "| * ) &) * * '} is given. We can then, in many different ways, form a sequence (x n ) which con- tains every term of each of these sequences and contains it exactly once. The proof consists simply in assigning a sequence (x n ) which fulfils what is required. For this purpose we write the given sequences in rows one be- low the other: r (t) ^(fc) ..(*) The "diagonal" of this system which joins the element x^ to the element x^ 1 then contains all elements x* for which A -f n = p, and no others. They are p 4- 1 in number. These terms we write down in succession, taking /> 0, 1, 2, . . ., and describe each of the diagonals say from bottom to top. Thus we obtain the sequence a:< > x (l) x (0} z (2) a: (1) a: (0) x< rE< 2) "'O ' ' 1 * * 1 *g * ' 1 ' * * * f which evidently fulfils the requirements. (Arrangement by diagonals*). Another arrangement frequently used is that "by squares". Here we first write the elements x^ , a;j p) , ..., x^ of the p ib row, then the elements standing vertically above x in the above system: as^"" 1 *, . . ., ad*. These groups of 2+ 1 terms are then written down in succession for p = 0, 1, 2, . . ., and this gives, beginning with :r (0) rr (1) x^ tf (0) x (2) x (2} x (2) x X M X (Q) ^0 * ' 1 1 *0 ' 1 2 ' 2 f 2 ' the arrangement by squares**. If some or all of the rows in the above system consist of only a finite number of terms, or if the system consists of only a finite number of rows, then the arrangements described above undergo slight and immediately ob vious modifications. * German: Anordnung nach Schrdglinien. (Tr.) ** German: Anordnung na<,h Quadraten. (Tr.) 10. Limiting points and upper and lower limits. 91 5. An example similar to 3. is the following: For every p^.2 there arc exactly />- 1 numbers of the form -r-H -- for which the sum of the positive / ft Wl integers k and m is equal to p. If we suppose these written down in succession, for p = 2, 3, 4, . . . , we obtain the sequence 33 4 4 j5 _5 ^ ' 2 ' 2"' 3 ' ' Y' 4' 6 ' G' " ' We find that this sequence has the limiting points 0, 1, --, ^-, -j-, . .. and no others. 6. As in the case of the limit of a convergent sequence, the limiting points of an arbitrary sequence may very well not belong to the sequence itself. Thus in 3. the irrational numbers, and m 5. the value 0, certainly do not belong to the sequence concerned. On the other hand, in both cases the value -J, for instance, is both a limiting point and a term of the sequence. We proceed to give a theorem which is fundamental for our purpose, due originally to B. Bolzano 10 , though its significance was first fully recognised by K. Weierstrass u . Theorem. Every bounded sequence possesses at least one limit- 54. ing point. Proof. We again determine the number in question by a suitable nest of intervals. By hypothesis there exists an interval / which contains all the terms of the given sequence (# tj ) To this interval we apply the method of successive bisection and designate as / x its left or right half according as the left half contains an infinite number of the terms of the sequence or not. By the same rule we designate a definite half of / t as / Q , and so on. Then the intervals of the nest (/J so formed all have the property that an infinite number of terms is contained in each, whilst to the left of their left endpoint there is always at most a finite number of points of the sequence. The point thus defined is obviously a limiting point; for if e > is given arbitrarily, choose from the succession of inter- vals J n one, say ] } , whose length is < K. The terms of (# M ), in number infinite, which belong to the interval / then lie ipso facto in the e- neighbourhood of , which proves all that we require. The similarity of the definitions of limiting point and limit (or limiting value) in spite of the difference emphasized in 53, 1 ("every limit is also a limiting point, but not conversely'') naturally creates a certain relationship between them. This is elucidated by the following 10 Rein analytischer Beweis des Lehrsatzes, dafi zwischcn je zwey Werthen, die em entgegengesetztes Resultat gewUhren, wenigstens eine reelle Wurzel der Gleichung liege, Prag 1817. 11 In Ins lectures. 92 Chapter II. Sequences of real numbers. 55. Theorem. Every limiting point of a sequence (a? n ) may be re garded as the limit of a suitable sub-sequence of (# n ). Proof. Since for every e > 0, we have, for an infinite number of indices, \x n || < e, we have, in particular, for a suitable n = k^, | x kl f | < 1; for a suitable n = 7e. 2 > A 1? we have similarly | x^ | < |, and in geneial, for a suitable n = k v > v -i 1 For the subsequence (x n ') == (x^ thus picked out, we have x n '+t;, as (xjt n |), by 26, 2, forms a null sequence. The proof of the theorem of Bolzano-Weierstrass gives occasion for a further most important remark: The intervals J n of the nest there constructed not only had the property that within them lay an infinite number of terms of the sequence (x n ), but as we noticed, they had the further property that to the left of the left cndpoint of any definite one of the intervals there lay always a finite number only of the terms of the sequence. From this, however, it follows at once that no further limiting point can lie to the left of the limiting point already determined. For if we choose any real number ' < , we have e = ^(f f ') < 0; choosing an interval J of length < e, we have the whole of the e- neighbourhood of the point ' lying to the left of the left endpoint of / and therefore containing only a finite number of terms of the sequence. Therefore no point ' to the left of f can be a limiting point of the sequence (#J, and we have the 56. % Theorem. Every bounded sequence has a well- defined least limit - * ing point (i. e. one farthest to the left). If we interchange right and left in these considerations, we obtain 12 quite similarly the 57. Theorem. Every bounded sequence has a well-defined greatest limiting point 13 (i. e. one farthest to the right). These two special limiting points we will designate by a special name. 58. Definition. The least limiting point of a (bounded) sequence will be called* its lower limit or Mines inferior. Denoting it by x, we write Hm re = x or lim inf x = x 12 Or by reflection at the origin. 13 These theorems are again obvious except in the case in which the sequence (x n ) has an infinite number of limiting points, like e. g. the sequence 53, 5. For among a finite number of values there must always be both a greatest and a least. * The German text has "untere Haufungsgrenze, unterer Limes, Limes inferior", (Tr.) 10. Limiting- points and upper and lower limits. 93 (possibly omitting the subscript n *<x>). // p, is the greatest li- miting point of the sequence, we write lirn x n = p, or lim sup x n = /* n-> n-> and call /t* the upper limit or limes superior of the sequence (# n ). We have necessarily always x^f*. Since every e- neighbourhood of the point contains an infinite number of terms of the sequence (zj, and since on the other hand only a finite number of terms of the sequence can lie to the left of the left endpoint of any such neighbourhood, K (or similarly fi) is also characterised by the following conditions: Theorem. The number x (or p) is the lower (or upper) limit of 59. the sequence (J if and only if, given an arbitrary e > 0, we have still for an infinite number of n's, x n < * + G ( or > f* ~~ e ) f but for at most a finite number 14 of n's, x n < x e (or > fi + z). Before we give a few examples and explanations of this theorem, let us complete our definitions for the case of unbounded sequences. Definitions. I. If a sequence is unbounded on the left, then we 60. will say that oo is a limiting point of the sequence ; and if it is unbounded on the right, we will say that -}-oo is a limiting point of the sequence. In these cases, however large we choose the number G > 0, the sequence has an infinity of terms 15 below G or above + G. 2. If therefore the sequence (x n ) is unbounded on the left, then oo is the least limiting point, so that we have to write x = lim x n = co . n->+oo Similarly we have to write fji =. lim x n = + oo if the sequence is unbounded on the right. In these cases, nowever large we choose the number G > 0, we have, for an infinity of indices, x n< G or x n>+ G - * The German text has "obere Haufungsgrenze, oberen Limes, Limes superior". (Tr.) 14 Or: There is an index n Q from and after which we never have x n < x e ( > /* + e) but beyond every index n, there is always another n for which x n < x + e <>*-). 15 Here therefore and similarly in the following definitions the portion of the straight line to the right of + G plays the part of an s-neighbourhood of + oo, the portion to the left of G that of an s-neighbourhood of oo. 94 Chapter II. Sequences of real numbers. 3. If, finally, the sequence is bounded on the left, but not on the right and (besides + oo j has no other limiting point, then -f- oo is not only its greatest, but at the same time its least limiting point, and we shall therefore equate the lower limit also to -f-oo: # = lima; n = +00; n->4-oo and correspondingly we shall have to equate the upper limit to oo, fj, lim x n = oo n->+co if the sequence is bounded on the right, but not on the left, and (besides oo) has no other limiting point. The former (latter) case occurs if and only if, given any G > 0, the inequality x n >G ( Xn< -G) holds for an infinite number of n j s, but the inequality x n <G (x n >~G) for at most a finite number of n's, that is to say therefore when x n -> + GO (-00), Cf. 63, Theorem 2. Examples and explanations. 61. 1. In consequence of the preceding definitions, every sequence of numbers now of itself defines, absolutely uniquely, two determinate symbols * and p } (which may now, it is true, stand f or -f- oo or oo , and which bear the re- lation x ~5 M to one another 16 . And the following examples show that * and n may actually assume all finite or infinite values compatible with ihe in equality x < |i. In fact, for the sequence we 1 iave 1. (n)=l,2, 3, 4, ... + 00 + 00 2. (f<- 1 >")EE + l f a + 2 l . + | l . + 4 f ... a + 00 3. a, b, a, 6, a, fc, . . . (a<b) a b 4. (a 4- t^")^*- 1, a-H-g , ---, + {-, a a 5. ((-!). *)==-!, 4-2, -3, +4,... GO 4-oo 6. (fl-V-'^ssa- I,- f .-|,-4 f ... 00 7. ( W)=E2 1, -2, -3,... 00 00 2. The reader should note particularly that it is not contradictory to theorem 59 that an infinite number of terms of the sequence should he to the left of x or to the right of p. Thus for instance we have, for the sequence ' ^e. for the sequence -2, +-J, ~i f +A, _ |. f ... evidently 18 We say of every real number that it is < + oo and > oo , and for this reason we occasionally designate it expressly as "finite". 10. Limiting points and upper and lower limits. 95 K s 1 , p s= -f- 1 > and both to the left of and to the right of fj, lies an infinite number of terms of the sequence (and between x and p lies no term of the sequence I). It is therefore not at all necessary that there should be only a finite number of terms of the sequence outside the interval * .../^. Theorem 59 only asserts in fact that at most a finite number of terms of the sequence can lie to the left of s or to the right of /w-f-e. 3. "A finite number of alterations" has no effect on the limiting points of a sequence none, in particular, on its upper and lower limits. These therefore represent an ultimate property of the sequence. 4. Since a sequence (x n ) determines both the numbers x and /w with complete uniqueness, and since their value, in connection with our definition, was also enclosed by a well defined nest of intervals, we have herein a new legi- timate means of defining (determining, giving) real numbers: a real number shall henceforth also be regarded as "given", if it is the upper or lower limit of a given sequence. This means of determining real numbers is evidently still more general than the one mentioned in 41, 1 since now the sequence utilised need not even be convergent, or be subject to any restriction whatever 17 . As may be seen, in the light of 55, we have also the following Theorem. The upper limit /i of the sequence (x n ), /j = lim x n , is 62. also, in the case JLI =j= oo, characterised by the two following conditions : a) the limit ' of every convergent sub-sequence (x n f ) of (a?J is invariably < yw ; but there exists b) at least one such stib- sequence, whose limit is equal to //; and correspondingly for the lower limit. A concept related to that of the upper and lower limits, though one which must be sharply distinguished from it, is the concept of upper and lower bounds of a sequence (# M ), which is derived from the following consideration: If no term of the sequence lies to the right of // = lima; n , so that for every n, #<[/*, then /i is a bound above (8, 4) of the sequence, but one which cannot be replaced by any smaller one; fi is therefore in this case the least bound above. But such a least bound also exists if there is a term of the sequence > p. For if for instance x is > /i, then by 59 there is certainly only a finite number of terms in the sequence which are ^> x p , and among these there is necessarily (8, 5) a largest one, say x . We then have, for every n, x n <^ x , i. e. x is a bound above of the se- quence, but again one, which cannot be replaced by any smaller one. Every sequence bounded on the right therefore possesses a definite least bound above. Since, in the same way, every sequence bounded 17 Whereas therefore a nest of intervals (with rational cndpoints) was at first to count as the only means of defining a real number, we have now deduced quite a series of other means which we now admit as equally legi- timate: Radix fractions, Dedekind sections, nests of intervals with arbitrary real endpoints, convergent sequences, upper and lower limits of a sequence In all these cases, however, we saw how at once to assign a nest ol intervals (with rational endpoints) which encloses the given number. 96 Chapter II. Sequences of real numbers. on the left must have a definite greatest bound below, we are justified in the following Definition. We define as the upper bound * of a sequence bounded on the right the least of its bounds above (invariably determinate by our pre- liminary remarks), and similarly as the lower bound * of a sequence bounded on the left the greatest of its bounds below. A sequence unbounded on the right is said to possess the upper bound + <x>, one unbounded on the left, to possess the lower bound oo . The concepts of upper and lower limits are due to A. L. Cauchy (Analyse alg<5hnque, p. 132. Paris 1821) but were first made generally known by P. du Bois- Reymond (Allgemeine Funktionentheorie, Tubingen 1882). Both nomenclature and notation have remained variable up to the present day. The particularly con- venient notation hm and hm used in the text was introduced by A. Pnngsheim (Sitzungsber. d. Akad. zu Munchen, vol. 28, p. 62. 1898), to whom the designations of upper and lower limits are also due **. It should be expressly pointed out again that the upper (and similarly the lower) bound is not necessarily determined by the tail-end of the sequence. Thus the upper bound of the sequence f - J is 1, and is obviously altered if the first term of the sequence is altered. The previous investigations of this paragraph were carried out quite independently of the considerations on convergence of 8 and 9, and give us, for this very reason, a new means of attacking the problem of convergence A of 9. It may be shewn that the knowledge of the lower and upper limits x and /x of a sequence the knowledge, therefore, of two numbers whose existence is a priori ensured entirely suffices to decide whether or how the sequence converges or diverges. We have in fact the theorems 63* Theorem 1 . The sequence (x n ) is convergent if and only if its lower and upper limits x and p are equal and finite. If A is the common value (different, therefore, from + GO or GO) of x and /z, then x n -> A. Proof, a) Let x = // and their common value =- A. Then, by 59, given e, there is at most a finite number of w's for which * German: Obere, untere Grenze (frontier). The word "frontier" is not usual in English writings, though sometimes found in French. The distinction between any bounds and the narrowest bounds is emphasized chiefly by the article the in the latter case; the upper bound and the lower bound always denoting the latter. For fear of ambiguity, however, the word "bound" in the general sense is avoided as much as possible in English text-books. (Tr.) ** We have omitted reference here to the untranslated term "Haufungsgrenze" of the German text: "Die im Texte benutzte ausfuhrlichere Bezeichnung Hdufungs- grerize soil nur den Unterschied zu der soeben defimerten unteren und oberen Grenze starker betonen". (Tr.) 10. Limiting points and upper and lower limits. 97 and similarly at most a finite number of #'s for which For every n ^ some w , we therefore have A < # n < A + , or | # n A | < e, i. e. the sequence is convergent and A is its limit. b) If, conversely, lim x n A, then, given e > 0, we have, for every n > n (s), A E < # n < A + Therefore the inequality # n <A + (>A-e) is satisfied for an infinite number of 's, but the inequality x n < A (> A + ) for at most a finite number of n's. The former inequalities (with <) imply Y. = A, the latter /z ~ A. This proves all that we required. Theorem 2. The sequence (x n ) is definitely divergent if, and only if, its upper and lower limits are equal, but have the common value 18 + oo or oo. In the former case it diverges to + oo, in the latter to oo. Proof, a) If x = ft + oo (or oo ), then this signifies, by 60, 2 and 3, that, given G > 0, we have from and after a certain w * n > + G (<-G); we therefore then have lim x n = + oo ( oo). b) If, conversely, lim x n ~ + oo, then, given G > 0, we have for every n after a certain // , x n > -f G; therefore the inequality x n < + G is satisfied for at most a finite number of n's, whereas the inequality x n > + G is satisfied for an infinite number of w's. But this implies, by 60, that x + oo and ipso facto also fj, = + oo. Therefore x /x + oo. And in precisely the same way we show that if lim x n = oo , then x ~ p = oo . From these two theorems we at once deduce further: Theorem 3. The sequence (x n ) is indefinitely divergent if and only if its upper and lower limits are distinct. The content of these three theorems provides us with the following Third main criterion for the convergence or divergence of a sequence: 64. The sequence (x n ) behaves definitely or indefinitely, according as its upper and lower limits are equal or distinct. In the case of definite behaviour, it is convergent or divergent, according as the common value of the upper and lower limits is finite or infinite. 18 In occasionally speaking of the symbols -f- and oo (which are cer- tainly not numbers) as "values", we make use of a mere verbal licence, to which no importance should be attached. 98 Chapter II. Sequences of real numbers. The following table gives a summary of possibilities as regards the convergence or divergence of a sequence and of the designations used in this connection. x = p , both = A -+- 00 x = // = -f OO or oo * o convergent (with limit A) lim y n H (n->+) *.-** (for n -> + 00} divergent (or possibly: con- vergent) towards (or: with limit) -f OO or oo; in both cases: definitely divergent. lim x n -f oo or oo , 4 ~>4-OO or OO indefinitely divergent convergent divergent definite behaviour indefinite behaviour 11. Infinite series, infinite products, and infinite continued fractions. A numerical sequence can be specified in the most diverse ways; this is sufficiently evident from the examples which have been given. In these, however, for the most part, the n th term x n was for conveni- ence given by an explicit formula, enabling us to calculate it at once. This is by no means the rule, however, in the applications of sequences in all parts of mathematics. On the contrary, the sequences to be examined generally present themselves indirectly. Besides several less important kinds, three types especially come into consideration; of these we will now give a brief discussion. 66. I. Infinite series. These are sequences given in the following way. A sequence is at first assigned in any manner (usually by direct indication of its terms), but without being intended itself to form the object of discussion. From it a new sequence is to be deduced, whose terms we now denote by s n , writing s o == a o> s i = a o + a \\ ^2 a o + a i + #2> and generally s n "= a o + #1 -1- a 2 -f . . . 4- a n (n = 0, 1, 2, . . .). It is the sequence (s n ) of these numbers which then forms the object of investigation. For this sequence (s n ) we use the symbolical expression ft 7. a) a 4- ^ 4 a 2 4 . . . 4 a n 4- . . . or more shortly or still more shortly and more expressively: n-O 11. Infinite series, infinite products, and infinite continued fractions. 99 and this new symbol we call an infinite series \ the numbers s n are called the partial sums or sections * of the series. We may therefore state the Definition. An infinite series is a symbol of the form 68. QC Za n or ~|-0 1 l +*2 + -" W--0 or 00 + a l + a 2 + f a n + ' by which is meant the sequence (s n ) of the partial sums s n - *o + i + + <*n (n = 0, 1, 2, . . .) \ Remarks and Examples. 1. The symbols 00 CO 00 fli -1 ^ " n ; -I- fli -I . . . -f tf m 4- a n M - M M j- 1 00 shall be entirely equivalent to Ea n . The index n is called the index of summation. w=- Of course any other letter may take its place GO CO -27 a v \ a {} -f i -f <** + The numbers a n are the term* of the series. They need not be indexed from on- wards. Thus the symbol 00 27 a A denotes the sequence (a lt a^ -f a 2 , a l -f a 2 + a 3t . . .) and more generally, denotes the sequence of numbers s p , s v+l , s P+Zt . . . given by s n = a p + a v+ 1 + + n for n ^ P,P -I- ! Here p may be any integer ~ 0. Finally we also write quite shortly a v when there is no ambiguity as to the values which the index of summation has to assume, or when this is a matter of indifference. 2. For H = 0, 1, 2, . . . let a n be e ) ^ ""; = (- v n > s) - (- i) n (2 + i); =4= 0, - I, ~ 2, ... * German : Teilsummen oder Abschnitte. ]00 Chapter II. ' Sequences of real numbers. We are then concerned with the infinite series > J o i Esl + 7 + T + + "' : o> 1 111 1T2 + 2^ + 3^4 + * " ? c ) i + i + i_j-... ; d) 0+1 + 2 + 3H ; ^ r ^ *"+ l ~ * ~~ ~2 + "if ~ 4~ ~^ ' ^(-1)^=1-1 + 1-1 + ; g) 1-3+5-7+9-. , ^ 1 1.1.1 And we have in these simply a new and as will be seen, very con venient symbol for the sequences (s , s , s a . . . .) for which s n is b ) ^TTo + oT^ + sTZ . 2 ^ 2 - 3 ^ 3 . 4 ^ (n + 1 ) (* + 2) n(n+l). c; = M + 1 ; d) = - 2 ; ( cf - 45 > 3 and 48,1); f) = H 1 - (-l) n+1 ] (see footnote 19); g) =(-l) n ("+ 1); "' ~ a (a + 1) T (a + 1) (a + 2) ^ ' ' ' ^ (a + n) (a + n + 1) /I 1 \ + /I 1_\ + _. + /. _ * * I -. 1 a a + n + 1" 3. We emphasise above all that the new symbols have no significance in them- selves. Addition, it is true, is a well-defined operation, always possible, with regard to two or any particular number of values, in one and only one way. The partial sums s n therefore, however the terms a n may be given, have under all circumstances definite values. But the symbol fa n has in itself no meaning whatever, not n-O even in a case as transparent, seemingly, as 2 a ; for the addition of an infinite number of terms is something quite undefined, something perfectly meaningless. It must be considered substantially as a convention that we are to take the new symbol to mean the sequence of its partial sums. lu Equal to 1 or 0, according as n is even or odd. 1 1 1 1 i 1 1 2' y 5' r 11' 13' 17' 1 i 1 i i 1 1 3' 7' 8' ir>' 24' 2<>' 31' 11. Infinite series, infinite products, and infinite continued fractions. 101 4. The reader should take particular care to distinguish a series from a se- quence 20 : A series is a new symbol for a sequence deducible by a definite rule from it. 5. The symbol with the sign of summation "JL can of course only be used when the terms of the series are formed by an explicitly assigned law, or when a particular notation is available for them. If for instance the numbers or the numbers are to be the terms of a series, we shall have to use the explicit symbols and 3 '" 7 + 8 + 15 + 24 + 2(3 + 3l + ' ' * and write down as many terms as necessary, till we may assume that the reader has recognised the law of formation. For the first of these two series, this may be expected after the term ^ : the terms arc the reciprocals of the successive prime numbers. In the second example it will not be known even after the term } f how to proceed : the denominators of the terms are meant to be the integers of the form P q - 1 (P,q=- 2, 3, 4, . . .) in order of magnitude. We now adopt the further convention that all expressions used to describe the behaviour, in respect of convergence, of a sequence are to be carried over from the sequence (s n ) to the infinite scries 2 a n itself. Thereby we obtain in particular the following Definition. An infinite series 2 a n is said to be convergent, definitely 69. divergent or indefinitely divergent, according as the sequence of its partial sums shows the behaviour indicated by those names. If, in the case of con- vergence, s n -> s, then we say that s is the value or the sum of the convergent infinite series and we write for brevity cr E a v = s, v -0 00 t so that a v denotes not only the sequence (s n ) of the partial sums, as laid down v~-0 in the preceding definition, but also the limit lim s n , when this exists 2i . In the case of definite divergence of (s n ), zve also say that the series is definitely divergent and that it diverges to + oo or oo according as s n -> + or -> oo. If finally, in the case of indefinite divergence of (s n ), Y. and p are the lower and upper limits of the sequence, then we also say that the series is indefinitely divergent and oscillates between the (lower and upper) limits Y. and fji. 20 The additional epithet of "infinite** may be omitted when obvious. 21 Exactly as we may now, in accordance with the footnote 9 to 41, 1, write 102 Chapter II. Sequences ot real numbers. Remarks and examples. 1. It is at once obvious that the series 68, 2 a, b and h converge and have for sums + 2, 1 and respectively; 2c and d are definitely divergent towards a + oo ; 2 e is convergent and has for sum the number s defined by the nest 2a ( s ve-i\ S2k)', 2 f , finally, oscillates between and 1, and 2g between oo and H-oo. 2. As regards the term sum the reader must be expressly cautioned about a possible misunderstanding: The number s is not a sum in any sense previously in use, but only the limit of an infinite sequence of sums', the equation 27 = s or a + a L -\ + H = s n-O is therefore neither more nor less than another way of writing lim s n = s or s n > s . It would therefore seem more appropriate to speak not of the sum but of the limit or value of the series. However the term "sum" has remained in use from the time when infinite series first appeared in mathematical science and when no one had a clear notion of the underlying limiting processes or, generally, of the "infinite" at all. 3. The number 5 is therefore no sum, but is only so named, for the sake of brevity. In particular, calculations involving series will in no wise obey all the rules for calculating with sums. Thus for instance in an (actual) sum we may introduce or omit brackets in any manner, so that for instance, 1 _ i + i _ i = (i _ i) + (i _ i) = i _ (i _ i) _ i = o. But on the contrary J; ( 1)"S3 1 1+1 1+ n=o is not the same thing as (1 - 1) + (1-1) + (1-1)+...= + + 0+ or as 1- (1-1) -(1-1) -(1-1) ==1-0-0-0 Nevertheless, calculations involving series will 'have many analogies with those involving (actual) sums. The existence of such an analogy has, however, in every particular case to be first established. 4. It is also, perhaps, not superfluous to remark that it is really quite 00 J paradoxical that an infinite series, say J5o~" should possess anything at all 22 1 . S that \< 5 a<*6<---; similarly from s 2jfe - gTr+r we deduce that 5 > 5 > 5 > Finallv *2k~~ $ 2k-i = ~^9TTT> if e> P ositi ve and tending to 0. By 46, 4 and 41, 5, we have s n ~ * (s^fc-i | 5 2Jt)' ^' ^ c ^ an< * ^j ** where these considerations are generalised, 11. Infinite series, infinite products, and infinite continued fractions. 103 capable of being called its sum. Let us interpret it in fourth- form fashion by shillings and pence: I give some one first 1 s. f then 1 / 2 s., then */ 4 s., then */ 8 s., and so on. If now I never come to an end with these gifts, the question arises, whether the fortune of the recipient must thereby necessarily increase beyond all bounds, or not. At first one has the feeling that the former must occur; for if I continue constantly adding something, the sum must it seems ulti- mately exceed every value. In the case under consideration this is not so, since for every n s n = 1 + 2 + 4 -f . . . + 2 7j , - 2 - 2n remains < 2. The total gift therefore never reaches even the amount of 2 s. And if we now, in spite of this, say that 2! 2 n ** equal to 2, then we are really only using an abbreviated expression for the fact that the sequence of partial sums tends to the limit 2. Cf. the well-known paradox of Achilles and the tortoise (Zenon's paradox). 5. In the case of definite divergence we can also, in an extended sense, speak of a sum of the series, which then has the "value" +00 or -co. Thus for instance the series is definitely divergent, and has the "sum" hoc, because by 46, .3 its partial 2S sums -* -f- oo . We write for short oo I n=l which is only another mode of writing for 6. In the case of an indefinitely divergent series however, the word "sum" loses all significance. If in this case litn s n x and lini s n fi (> x), then we said, in the above, that the series oscillates between x and, //. But it must be carefully noted (cf. 61, 2), that this refers only to a description of the ultimate behaviour of the series. In fact the partial suras s n need not lie between x and p. Thus, for instance, if a == 2, and for w]>0, \ve can at once verify that * = + + + = (- i)"jqrj ( = o, 1,2, ...) and therefore lim s n = 1 , lim s n = -f- 1 . But all the terms of the sequence (s n ) 2a If therefore the payments discussed in 4. have the values 1 s., l / s., */ 8 s., l / 4 s.,... the fortune of the recipient now does increase beyond all bounds. It is not at first at all obvious to what it is due that in the case 4, the sum does not exceed a modest amount, whereas in the present case it exceeds every bound. The divergence of this series was discovered by John_J$ejyuiuUL^ and published by James Bernoulli in 1689; but seems to have been already known to Leibniz m 1673. 11. Infinite series, infinite products, and infinite continued fractions. 105 they must be taken, in a precisely similar manner to the infinite series just considered, simply as a new symbolic form for the well-defined sequence of the partial products />i = r, Pz -^ u\ " 2 ; ; /> = *i 2 M ; - - However we shall later, with reference to the exceptional part played by the number in multiplication, have to make a few special conventions in this connection. 1. If for instance we have, for every n ^ 1, n n - -, - ., then the infinite product fr > + 1 ) 2 22 32 42 52 ( n i- ! ) 2 , n~(iT+ 2) r 1 3 '2-4* 3 5 4 - ' ' ' n (n |- 2) ' ' ' n -- L represents the sequence of numbers 4 2-3 2-4. 2(w |- 1) Pi = %', Pz = - 4 ~, />3 = - r> - ..... />M ~~ w ~:p~2 ~ - - 2. The additions and remarks just made in I retain mutatis mutandis their significance here. All further details will be considered later (Chapter VII). in. Infinite continued fractious. Here the sequence (v w ) under examination is formed by means of two other sequences (,, 2 . . .) and (6 , b lt . . ), by writing: #0 - - 'V and so on, x n , in the general case, being deduced from x n ,_ l by substituting for the last denominator & n _ t of .v n _, the value b n _ l -f , ", and proceeding thus ad infitntum. For the "infinite continued fraction" so formed the notation is fairly usual. The most natural notation for it would be 71-1 Here also a few special conventions have to be made, to take the fact into account that in division the number again plays an exceptional part. The subject of con- tinued fractions we shall not, however, enter into in this treatise 24 . Of the three modes of assigning a sequence discussed above, that by infinite series is by far the most important for all applications in higher mathematics. We shall therefore have to deal mainly with these. Since series merely represent sequences, the introductory developments of 9 provide us with the points of view from which a given series will have to be investigated: Together with the problem A which concerns the convergence or divergence of a given series, we have again the harder problem B 9 which relates to the sum of a series already seen to be convergent. And for exactly the same 24 A complete account of their theory and applications is given by O. Perron, Die Lehre von den Kettenbruchen, 2 nd Edition, Leipzig 1929. 106 Chapter II. Sequences of real numbers. reasons as we there explained, the second problem will generally present itself in the form: A series 2 a n is known to be convergent; does its sum coincide with that of any other series or with the limit of any other sequence, or does it stand in any assignable relation tb such another sum or limit? 2B Since the problem A is the easier and since in contradistinction to problem B it admits of a methodical solution, we will proceed in the first place to give our attention to this in detail. Exercises on Chapter II 26 . 9. Prove Theorems 15 to 19 of Chapter I by the method indicated in the footnote to 14. 10. Prove in all details that the ordered arrangement, defined by 14 and 15, of the system of all nests of intervals, obeys each of the theorems of order 1. (For this cf. 14, 4 and 15, 2.) 11. Carry out the details of the proof required on p. 32; i, e. prove that the four modes of combining nests of intervals, defined by 16 to 19, obey all the fundamental laws 2. 12. For fixed 3, with 2 < 1, 13. For arbitrary positive a and /?, (loglostt)"^^ (log nf Vs~ * 14. Which of the two numbers (--} and f ^~2j 2 is the larger? 25 Thus e.g. the series l + l-f +-.+ . ..-J r+"* will easily be 6\ o 1 nl shown to converge. How do we see that its sum coincides with the number * / \\n given by the sequence I 1 -| -- 1 ? Similarly we may very soon convince our- \ n J selves of the convergence of the two series l + l + -+... + +... and 1-.. + ..-. + -.... O But how do we discover that if s and s' are their sums, $ = ---s' 9 and 4s' = ;r o (i. e. equal to the limit in a third limiting process, which occurs in relation to the circle; cf pp. 200 and 214)? 20 In several of the following exercises, a few of the simplest results with regard to logarithms, and the numbers e and yi r are Assumed known, although they are only deduced later on in the text. Exercises on Chapter 11. 107 15. Prove the following limiting relations: r * Ln-f- Note that in examples a) to d) a term by term passage to the limit gives a wrong result, whereas in e) it gives a correct result 16. Let a be >0, x l > and the sequence (X L) x^ t . . .) defined by the convention that for n ft) b) ' Shew that in case a) the sequence tends monotonely to the positive root of jc^ x a = 0; that in case b) it tends to that of x* + x a = 0, but with x n lying alternately to the left and to the right of the limit 17. Investigate the convergence or divergence of the following sequences' a) X Q , Xi arbitrary; for every n>2, # n = j- (&n-i H-a-a)t b) X Q , x it . . ., x p -. t arbitrary; for every n > p ,-,+ + Xn-p 1 *i a ai * a p given constants, e. g. all equal to ~ j, c) x Q1 x l positive; for every w>2, x n ~ \x n ^ l a? w - 3 ; 2:r .r d) x , XL arbitrary; for every n^2, x n ^-~* '*~ 9 -. 18. If in Ex. 17, c we put, in particular, # = 1, a? 1 = 2, then the limit of 8 the sequence is = "y/ 4 . 19. Let a lt a a , . . ., a p be arbitrary given positive quantities and let us write, for n = 1, 2, . . . ^ ^ 1^ -* = s n and Vs7 = a; M . J03 Chapter IT Sequences of real numbers. Show that x lt always increases monotonely and if one, say a lf of the given numbers is greater than all the others, then x n -> a as limit. (Hint: First show that 20. Somewhat similarly to last Ex., write n . n _ n t __ * - " = s n ' and (s n ') = x/ and show that a:,/ decreases monotonely and -* ya, a. 2 ... a p . 21. Divide the interval a ... & (0 << a < 6) into M equal parts; let or = a ( Xj, x 9 , . . ., x n = b denote the points of division. Show that the geometric mean . . . . n -I- 1 b IL a and the harmonic mean -^ ^ 1 "^ log 6 log a" 22. Show that in the case of the general sequence of Ex 5 - "(-/*)" JJ. Set a;>^0 and let the sequence (.r n ) be defined by For what values of x is the sequence convergent? (Answer: If and only if 1 24. Let lim# n = *, hm =/*, Hm x n ' x', \}mx n ' = f/. What may be said of the position of the limits for the sequences Discuss all possible cases. 25. Let (a w ) be bounded and (with the possible exception of a few initial terms) let us put Then (a n ) and (/?) have the same upper and lower limits. The same holds if we put . _ _. n nlogn/ n nlogn 26. Does Theorem 43, 3 still hold if 9=0 or = + oo? 27. If the sequences (x n ) and (y n ) given in 43, 2 and 3 are monotone, then so are the sequences (#') and (y n ') mentioned there. Exercises on Chapter II. 109 28* If the sequence (-j~j is monotone and & n >>0, then the sequence having w th term H is also monotone. 29. We have provided the limit on the right exists and (a,,) and (&) are null sequences, with (fc n ) monotone. 3O. For positive, monotone c n 's, XI H ----- \~Xn t implies c a? + c t x l H ----- h c n .r n _^ provided * is bounded and C n ~> -f oo . (Here C n = c -f c t -f- -f c n .) \ ts n / 31. If 6 n >0, and 6 + ^H ---- +& = ^-> + oo, and a^-^ + oo, then i " (v n +i ~ X n ) ->| 6 n implies tftp * I- &i * t + ' ' + ^n Vr"| __ v > * nfi *o+ir+--- + ^n J ^ 32. For every sequence (# n ), we invariably have (Cf. Theorem 161.) 33. Show that if the coefficients a^ t of the Theorem of Toeplitz 43, 5 are positive, then for tft/ery sequence (x n ) the relation lim x n ^ lim .T,/ < hm x n holds, where a?/ = a n o; + a n ^ + ---- hn*- Part II. Foundations of the theory of infinite series. Chapter III. Series of positive terms. 12. The first principal criterion and the two comparison tests. In this chapter we shall be concerned exclusively with series, all of whose terms are positive or at least non-negative numbers. If 2 a n is such a series, which we shall designate for brevity as a series of positive terms, then, since a n ^ 0, we have s n = *n-l + <*n ^ s n~l> so that the sequence (s n ) of partial sums is a monotone increasing sequence. Its behaviour is therefore particularly simple, since it is then determined by the first main criterion 46. This at once provides the following simple Jd fundamental First principal criterion. A series with positive terms either con- verges or else diverges to + <x> . And it is convergent if, and only if, its partial sums are bounded l . Before indicating the first applications of this fundamental theorem, we may facilitate its use by the following additional propositions: Theorem 1. If p is any positive integer, then the two series V) TO S a n and 27 a n 7i n=/> converge and diverge together 2 , and when both series converge, 1 Only boundcdness on the right (boundcdness above) comes into question, since an increasing sequence is invariably bounded on the left. 2 More shortly: We "may" omit an arbitrary initial portion. For this reason, it is often unnecessary to indicate the limits of summation (between which the index n is made to vary). 110 12. The first principal criterion and the two comparison tests. Ill Proof. If s n (n = 0, J, . . .) are the partial sums of the first series, and s n f (n ~ />,/> + 1, . . .) those of the second, then, for n^p, s n ^<*o -h !+- + *-i + s n ', whence, for ->oo, both statements follow, even without requiring the terms a n to be non-negative. Theorem 2. If Zc n is a convergent series with positive terms, then so is E y n c ny if the factors y n are any positive, but bounded, numbers 3 . Proof. If the partial sums of 2c n remain constantly <C K and the factors y n < y, then the partial sums of 2y n c n obviously remain always < y K, which, by the fundamental criterion, proves the theorem. Theorem 3. // 2d n is a divergent series with positive terms, then so is 2d n d n , if the factors d n are any numbers with a positive lower bound d. Proof. If G > be arbitrarily chosen, then by hypothesis the partial sums of 2 d n , from a suitable index onwards, are all > G:d. From the same index onwards, the partial sums of 2d n d n are then > G. Thus 2& n d n is divergent. Both theorems are substantially contained in the following Theorem 4. // the factors a n satisfy the inequalities then the two series with positive terms 2 a n and 2a n a n converge and diverge together. Or otherwise expressed. Two scries with positive terms a n and 2a n ' converge and diverge together if two positive numbers a' and a" can be\assigned for which, constantly, (or at least Irom some n onwards) 4 ct < < <*>" a n in particular therefore if a n ' ~ a n or, a fortiori, if a n f * a n (v. 40, 5). Examples and Remarks. 71. 1. If K is a bound above for the pattial sums of the series 2a n with positive terms, then the sum 5 of this series is < K (v. 46, 1). 2. The geometric series. Given a> 0, and the so-called geometric series 00 n=0 wu have, if a S> 1 1 then s n > and so (s n ) is certainly not bounded; the series 8 We shall in future usually denote by c n the terms of a series assumed convergent, and by d n those of a series assumed divergent. 4 Since, in this formulation of the hypotheses, division by a n occurs, the assumption is of course implied that a n > and never =0. Corresponding restrictions should be observed in the more frequent cases in the sequel. 112 Chapter III. Series of positive terms, is therefore in that case divergent. But if a << 1 , then 1 a"* 1 s n = 1 -f a + a' 2 -f -f a* = ^ , (cf . p. 22, footnote 13) and therefore we have, for every n, . 1 so that the series is then convergent. Since further 1 l-a 1-a forms a null sequence, by 1O, 7 and 26, 1, we at the same time obtain this is rarely the case a simple expression for the sum of the series: cr> 00 1 111 3. The series y] - --r-r === - ^ 4- ^r- -f ^- - -f- has the partial sums J^TI n (n -f- I) I & o o*4r " These are constantly < 1 , the scries is therefore convergent. As it happens, we can see at once that s n + 1 , so that s = 1 . / 1 1 1 / v 4. Harmonic series. 2, 1 + ?H H --- h is divergent, for, ^ -- - -- n=sl n 4 n as we saw in 46, 3, its partial sums diverge 5 to + QO. But the series V - 1 + - l is convergent. For its n th partial sum is =sl 1 . 1 . .1 t hence and therefore ^ w is constantly < 2, so that the given series is convergent. The sum s is not so readily obtainable in this case; we have however at any rate s < 2, indeed certainly * < . We shall find later (see 136, 156, 189 and 210) that s = ^-. A series of the form 2 ~~^ is called an harmonic series. 00 1 11 6. The series 2 ~ s * + 1 + 7 + o~i + nas the partial sums s n=on\ 1 61 Si = 2, and for n ^ 2, 6 Cf. footnote 2H, p. 103. 12. The first principal criterion and the two comparison tests. 113 Replacing each factor in the denominators by the least, namely 2, we deduce that ^ 9 1 1 _ J Tl " ^ O " O . O l~ * ~T" t) rt n & * & M Z . . . A = 2 + i + + ... + 2^1 = 3 - 2 ^ < 3. The series is therefore convergent, with sum ^ 3. We shall see later that this sum coincides with the limit e of the numbers ( 1 4- j . 6. As we remarked above that every series with positive terms represents a monotone increasing sequence, so we see, conversely, that every monotone in- creasing sequence (#, x lt . . .) may be expressed as a series with positive terms, provided x is positive. We need only write for, actually, and all the a n 's are T 0. From our fundamental theorem we shall in due course deduce criteria which are more special, but are also easier to manipulate. This we shall be enabled to do chiefly by the instrumentality of the two following "com- parison tests' ' *: Comparison test ofjthe_l^_hind. 72. Let 2c n and 2d n be two series zvith positive terms, already known to he the first convergent, the second divergent. If the terms of a given series 2 a n , also with positive tertns, satisfy, for every n > a certain m, a) the condition <*>n ^ Cfi. then the series 2 a n is also convergent. //, however, for every n> a cer- tain m, b) we have constantly then the series 2 a n must also diverge 6 . Proof. By 70, 1, it suffices to establish the convergence or di- r/5 vergence of 2 a n . In case a) the convergence of this series results oo at once, by 70, 2, from that of 2 c n , because by hypothesis we may, * German: Vergleichskriterien. (Tr.) fl Gauss used this criterion in 1812 (v. Werke III, p. 140). It was not, how- ever, formulated explicitly, nor was the following test of the 2 nd kind, before Cauchy, Analyse algbrique (Pans 1821). 114 Chapter III. Series of positive terms. for every n > m, write a n *= y n c n , with y n <^l. In case bl the di- CO vergence results similarly 7 from that of d n , because here we may n=tn+l write a n = (5 n d n , with $ w ^> 1. 73. Comparison test of the 2 nd kind. Let c n and 2 d n again denote respectively a convergent and a divergent series of positive terms. If the terms of a given series a l of positive terms satisfy, for every n^> a certain m, a) the conditions then the series 2a n is also convergent. If, however, for every n^> a certain m> we have b) constantly then 2 a n must also diverge. Proof. In case a), we have for every The sequence of the ratio y n = is, from a certain point on- wards, monotone descending, and consequently, since all its terms are positive, it is necessarily bounded Theorem 70, 2 now establishes the convergence. In case b) we have, analogously, 9 ~^ ]> ~ , so that the "+i a * ratios <J n = increase monotonely from a point onwards. But as they are constantly positive, they then have a positive lower bound. Theo- rem 70, 3 now proves the divergence. These comparison tests or criteria can of course only be useful to us if we are already acquainted with a large number of convergent and dhergent series with positive terms. We shall therefore have to lay in as large a stock as possible, so to speak, of series whose con- vergence or divergence is known. For this purpose the following examples may form a nucleus: 7 Or else almost more concisely : In case a) every bound above of the partial sums of 2 c n is also one for the partial sums of Sa n \ and in case b), the partial sums of a n must ultimately exceed every bound, since those of 2d n do so. 12. The first principal criterion and the two comparison tests. 115 Examples. 1. was seen to be divergent, 2 3 convergent. By the first comparison 74. test, the so-called harmonic series !,' is therefore certainly divergent for a ^ 1, convergent for a ^ 2. It is, however, only known in the case a ---= even integer how its sum may be related to numbers occurring in other connections; for instance we shall see later on that for a 4 . 7T 4 the sum is ^. 2. By the preceding, the convergence or divergence of 27 - only remains questionable in case 1 < a < 2. We may prove as follows that the Aeries converges for every a * 1 : To obtain a bound above for any partial sum s n of the series, choose k so large that 2 k > n. Then Here we group in one parenthesis those terms whose indices run from a power of 2 (inclusive) to the next power of 2 (exclusive). Replace, in each pair of paren- theses, every separate term by the first; this involves an increase ot value and we have therefore 2 4 2 A -- 1 *^ 1 + 2 + 4 + + (>-.) If we now write for brevity r,^ ^, a positive number certainly < 1, since a > 1, then we have and since this holds for every w, the partial sums of our series are bounded, and the series itself is convergent, q. e. d. (Cf. 77.) All harmonic series 2 "^for a ^ 1 are divergent, and for a > 1, convergent. In these, with the geometric series, we have already quite a useful stock of com- parison scries. 3. Series of the type where a and b are given positive numbers, also diverge for a < 1, converge for a > 1. For since u wchave 5 and 70. 4 Droves the truth of our statement. 116 Chapter III. Series of positive terms. Accordingly the series 11 ** 1 1 + 3 i + o + a n z _ (2^nr in particular, are convergent for a > 1, divergent for a 5^ 1. r> 4. If 2 c n is a convergent series with positive terms, and we deduce from n -o it a new series c n ' by omitting any (possibly an infinite number) of its terms, or by inserting in any way terms with the value 0, thus "diluting" the series, then the resulting "sub-series" 2 c n f is also convergent. For every number which is a bound above for the partial sums of S c n is then also a bound above for those of the new series. In accordance with this, the series 2 -&> where p runs through all prime in- tegral values, i. e. the series * 1,1,1 ,1,J_, 2<x i" ga i" ^a T 7<x i H<x ~r is certainly convergent for cc > 1 . (On the other hand, of course, we cannot conclude without further examination that it diverges for a. < 1 !) 5. Since 2 a n is already recognised as convergent for < a -< 1 , we infer in particular the convergence of V 1 -1 , _!_- 4-JLx n ^i io ~ 10 "*" io a "*"""*" IO B "*"""' If z l9 *$, > *> denote any "digits", i. e. if each of them be one of the numbers 0, 1, 2, . . ., 9, and if * is any integer ^-0, then, by 7O, 2, the series is also convergent. Thus we see that an infinite decimal fraction may also be regarded as an infinite series. In this sense we may say that every infinite decimal fraction is convergent and therefore represents a definite real number. In this form of series we also have, according to our customary order of ideas, an immediate conception of the value of its sum. / 13. The root test and the ratio test. We prepare the way for a more systematic use of these two comparison tests, by the two following theorems. If we take as com- parison series, to begin with, the geometric series 2a n , with 0<0<1, then we immediately obtain the 75. Theorem 1. //, given a series 2 a n of positive terms, we have, from some place onwards in the series, a n <^ a n with < a < 1, i. e. 13. The root test and the ratio test. 1 ] 7 then the series is convergent. It however, from some place onwards* then the series is divergent. (Cauchy's root test 8 .) Supplementary note. For divergence it clearly suffices that }/~a^>.\ should be known to hold for infinitely many distinct values of n. For we then also have, for those values of n, a n ^ 1; and a particular partial sum s m will consequently exceed a given (positive integral) number G, if m is chosen so large that the inequality a n > 1 occurs at least G times while < n < m . The sequence (s n ) is therefore certainly not bounded. The second comparison test gives immediately: Theorem 2. //, from some place onwards in the series, a n > 0, and then the series 2a n is convergent. If however, from some place onwards 9 then the series 2 a n is divergent. (Cauchy's ratio test 9 .) Remarks and Examples. 70* n _ 1. In both these theorems, it is essential for convergence that ^a n and - M+1 respectively should be ultimately lens than a fixed proper fraction a. It does not at all suffice for convergence that we should have V^<1, or ~^<1 for every n. An example presents itself at once in the harmonic series for which we certainly always have 1 and also -^--l^l -- <1, though the series diverges. It is quite essential that the root and ratio should not approach arbitrarily near to 1, 2. If one of the sequences ( V^nJ or f ?"-!J is convergent, say with limit , then theorems 1 and 2 show that the series 27 a * is convergent if <!, 8 Analyse algbrique, p. 132 seqq. 9 Analyse alggbrique, p. 134 seqq. 118 Chapter III. Series of positive terms. n 1 -- a divergent if a > 1. For suppose y a n -> a <C 1 , for instance; then e = - ;> t and m may be determined so that, for every n>w, we have n . < <* -1- = -- = a . And since this value a is < 1 , theorem 1 proves the convergence. If on the contrary a > 1 , then g' every w > w', we have contrary >!, then g' = ^ }> 0, and m' may be so determined that, for 2 And since this value a is >> 1, theorem 1 proves the divergence. The proof in the case of the ratio is quite analogous. If a=l, these two theorems prove nothing. 3. The reasoning just applied in 2. is obviously also legitimate when llm "y/a n or lim -^^ is < 1, in the one case, and lira ya n or lim --^^ is > 1, a n a n in the other. If one of these upper or lower limits is =1, or the upper limit > 1, the lower < 1, then we can infer almost nothing as to the convergence or diver- gence of Z a n . The supplementary note to 75, 1, however, shows that, in the root test, it is sufficient for divergence 10 that "lim ^/a n > 1. 4. The remarks just made in 2. and 3. are so obvious that, in similar cases in future, we shall not specially mention them. 5. The root and ratio tests are by far the most important tests used in practice. For most of the series which occur in applications, the question of convergence or divergence can be solved by their means. We append a few examples, in which x, for the present, represents a positive number. a) 2n a x n (a arbitrary). Here we have ?st_(!Ll)., .*,, a n \ n J f as !Lzt__ = 1 ^ --- * 1 and is permanently positive (v. 38, 8). The series is n n therefore and this without reference to the value of a convergent if *< 1, divergent if x> 1. For x = 1 our two tests are inconclusive; however we then get the harmonic series, with which we are already acquainted. 2 n=o \ n / n=o Here we have 10 Thereby the criterion obtains* a disjunctive form. Sa n is convergent iverj and 42.) n or divergent according as limya n is <; 1 or >> 1. (Further details in 36 13. The root test and the ratio test. 119 Hence this series too is convergent for x < 1, divergent for x > 1, whatever be the value of p. For x = 1 and p ^ it obviously diverges, since then w -- 1 ^ 1 for every n. In the case of convergence we shall later on find for its sum the value - * Here we have for every x > the series is therefore convergent for every x ^> 0. For the sum we shall later on find the value e x . d) / n i^ is convergent for #>0, as I/ = >0. M fe ^ n n n e) J^ 1 -- , n is convergent 11 , as again \/a n -> 0. f) A'r convergent, because n <^; ^ convergent, because a n = -- L1 -' < -^ for every n>2; divergent> becausc an V _._ . convergent, because a n << - . ^ v 'n(f+n*) . n^ g) 27..- vp (/> fixed > 0), is divergent, since by 38, 4 from some n on- wards (log n) p < n. h) 2j-. - y^- n is convergent, as we may at once recognize by writing the generic term in the form 1 11 In this series, summation may only begin with n = 2, since log 1 0. Such and similar obvious restrictions we shall in future not always expressly men- tion; it suffices, for the question of convergence or divergence, that the indicated terms of the series, from some place onwards, have determinate values. In all that follows, as already agreed on p. 83, the sign "log" will always stand for the natural logarithm, i. e. that to the base e (46 a). ] 20 Chapter III. Series of positive terms On the other hand is divergent, because by 38,4 and Ex. 13, (log log nf < log n from some n onwards, so that the generic term of the series is >> . 14. Series of positive, monotone decreasing terms. Before passing from these quite elementary considerations, we will mention a particularly simple class of series of positive terms, namely those series whose terms a n , at least from some place onwards, form a monotone sequence. To this class belong nearly all the series given as examples above and also the majority of those which occur in applications. For such series we have the following: 00 77. Cauchy's theorem of convergence 12 . // a n is a series whose n=i terms form a positive monotone decreasing sequence (a n \ then it con- verges and diverges with ^2 fc a 9 k^a JL + 2a a + 4 4 -f Sa s + *=o Preliminary remark. What is particularly remarkable in this theorem is that it shows that a small proportion of all the terms of the series suffices to determine the convergence or divergence of the whole series. For this reason it is also called the condensation theorem. It shows that the harmonic series J5J , for instance, is certainly diver- gent, for it converges and diverges with the series which is unmistakably divergent. And speaking generally, the series is n a inferred to converge and diverge with the series t* but this is a geometric series and therefore converges or diverges according as a ;> 1 or a < 1 . These examples also show us that the convergence or divergence of 22 fl 8 ft is often more easily ascertained than that of the series 2 a n itself; it is just in this that the value of the theorem lies. Proof. We denote the partial sums of the given series by s n , those of the new series by t k . Then we have (cf. 74, 2) 12 Analyse algbrique, p. 135. 14. Series of positive, monotone decreasing terms. 121 a) for n < 2 k i. e. b) for n > 2 . e. Inequality a) shows that the sequence (s w ) is bounded if the sequence (t^ is bounded; inequality b), conversely, that if (s n ) is bounded, so is (t k ). The two sequences are therefore either both bounded or both un- bounded, and therefore the two series under consideration either both converge or both diverge, q. e. d. Before given further examples illustrating this theorem, we may extend it somewhat 13 ; for it is immediately evident that the number 2 plays no essential part in the theorem. In fact we have, more generally, the Theorem. // Sa n is again a series whose terms form a positive 78. monotone decreasing sequence (aj, and if (g , g lf . ..) is any monotone increasing sequence of integers, then the two series CO 00 n=0 n *=0 k ff k are either both convergent or both divergent, provided g k , for every k > , fulfils the conditions grc > gfc-i ^ and &+! Sic^ M ' fefc ~ &c-i) in the second of which M stands for a positive constant. Proof. Exactly as before we have a) for n < g k , denoting by A the sum of the terms possibly preceding a ffo (or otherwise 0), ^ A + (g go) a Qo -f- + (fifc+i 6W a g k > i. e. 18 Schlomilch. O.: Zeitschr. f. Math. u. Phys., Vol. 18, p. 425. 1873. 14 The second condition signifies that the gaps in the sequence (&), re- latively to the sequence of all positive integers, must not increase at too great a rate. 122 Chapter III. Series of positive terms. b) for n > g k *. ^ f fc > (*+H ----- M,,H ----- K'Vt+iH ----- h%) ^ fei - go) f , H ----- H (& - fo-i) V ^ (&, - giK, H ----- h fewi &) fc ^ t t -t . And from the two inequalities the statements in question follow in the same way as before. 79* Remarks. 1. It suffices of course that the conditions in either theorem be fulfilled from and after a definite place in the series. Therefore we may, in the extended theorem, suppose, as a particular case, ft _8*. =4*,..., or =[g*] where g is any real number >> 1 and [g H ] the largest integer not greater than g*. We also satisfy the requirements of this theorem by taking & = **, =*', =*,... . 00 With gfrssh* we obtain, for instance, the theorem that the series 22 a m >f n=o (a n ) is a positive monotone decreasing sequence, converges and diverges with + 7a tf -f We may also replace this last series, according to 70, 4, by the series 2. J^ 7 : - is divergent, although its terms are materially less than those of the harmonic series; for according to our theorem, this series con- verges and diverges with and is therefore, by 70, 2, like the harmonic series, divergent. The divergence of this series and of those considered in the next examples was first discovered by N. H. Abel 16 (v. CEuvres II, p. 200). co 1 3. 5? = - : - : - is also still divergent, although its terms are again =3 * 10gtl. lOg lOg tt 6,6 6 considerably less than those of the Abel's series just considered. For by Cauchy's theorem it converges and diverges with 2*. log 2*. log (log 2*) *S*log2.1og(*log2)' 16 Niels Henrik Abel, born Aug. 5 th , 1802, at Findoe near Stavanger (Nor- way), died April 6 th , 1829, at the Froland ironworks, near Arendal. 14. Series of positive, monotone decreasing terms. 123 and this, since log 2 < 1 , has larger terms than Abel's series 5? - discussed k log k above, and must therefore diverge. 4. Thus we may continue as long as we please. To abbreviate, let us denote by log r x the yP le repeated or iterated logarithm of a positive number x, so that Iog x = x > Io i = log , log a x = log (log x), ... log r x = log (log,..! x) . We may also take log_ t x to denote the value e x '. These iterated logarithms only have a meaning if x is sufficiently large; thus loga; only for a;>0, Iog 2 x only for sc>l, Iog 3 x only for x>e, and so on; and we shall only place them in the denominators of the terms of our series if they are positive, i. e. log x only for x> 1, log a x only for x>e, Iog 8 a; onlv * or x > e * > and so on - H therefore we wish to consider the series og*... log, n then the summation must only begin with a suitably large index, whose exact value, however, (by 7O, 1), does not matter. Since the logarithms increase monotonely with w, and the terms therefore decrease monotonely, the series, by Caucfry's theorem, converges and diverges with ^ 1 and this, since 2 <e t must certainly diverge, if _ k log k . . . log^ x k diverges. Since the divergence of the latter series was proved for p = 1 (and = 2), it follows by Mathematical Induction (2, V) that it diverges for every p>l. 5. The series above considered, however, become convergent if we raise the last factor in the denominator to a power > 1. That converges for n a 1 , we already know. If we assume proved for a particular (integer) p > 1, that the series 16 /*\ \p __ ~ _ / ^ j\ a ) is convergent, it follows just as before that the series n-logn ... log^n-flog^n) 01 is also convergent. For this, by the extended Cauchy's theorem 78, converges and diverges with the series we choose gfc = 3* p 3* +1 -3* * 3* log 3*... (log, 8*)"' ' For p = 1 , this reduces to the series 124 Chapter III. Series of positive terms As 3>, this series has its terms less than those of the series (*) (assumed convergent), if the terms of the latter are multiplied by 2 (which by 70, 2 leaves the convergence undisturbed). The series brought forward in the two last examples will later on render us most valuable services as comparison series. We will prove one more remarkable theorem on series of positive monotone decreasing terms, although it anticipates to a certain extent the general considerations on convergence of the following chapter (v. 82, Theorem 1). 80. +j Theorem. // the series 2 a n of positive monotone decreasing terms is to converge, then we must have not only a n > 0, but 17 n a n -* . Proof. By hypothesis, the sequence of partial sums a -f- a \ ~H -J- a n = s n is convergent. Having chosen e > 0, we can therefore so choose m that for every v > m and every i ^> 1 we have | s v+;i s v | < -* , i. e. If we now choose n > 2m, then, taking v = [^n], the largest integer not greater than n, we have v^m and therefore flv+i + a v+2 H ----- h < y; a fortiori, therefore, (n -)*< ~ and Therefore na n +Q, q. e. d. Remark. We must expressly emphasize the fact that the condition n a n > is only a necessary t not a sufficient one for the convergence of our present type of series, i. e. if n a n does not tend to 0, then the series in question is certainly divergent 18 , while n a n * does not necessarily imply anything as to the possible convergence of the series. In point of fact, the Abel's series ^ - - diverges, although it has monotone decreasing terms and na n - logn Olivier, L.: Journ. f d. reine u. angew. Math., Vol. 2, p. 84. 1827. 18 Accordingly, the harmonic series ^ , for instance, must diverge n because it has monotone decreasing terms, but n- - does not tend to 0. Exercises on Chapter 111. 125 Exercises on Chapter III. 34. Investigate the behaviour (convergence or divergence) of a series a n , for which a n , from some index onwards, has the following values: _wl ^% + tt\ n" ' \ n )> V /n + * - 35. If 2 d n diverges, so also does -? (, B >0). * What is tne behaviour of 36. Under the same assumption that d n diverges and d n >Q t what is * the behaviour of the series ,? .,, 1 + a a 37. Suppose /> n -* + oo . What is the behaviour of the series 2 Pn n> ^'. 38. Suppose p n * -\- > , but with 1 < llm (p n +L p n ) What must be the upper and lower limits of the sequence () so that converge or so that it diverge? 39. For every n > 1, 4O. The sequence of numbers is monotone descending. 41. If 2a n has positive terms and is convergent, then 2 ^a n a n+1 is also convergent. Show by an example that the converse of this theorem is not true in general, and prove that it does nevertheless hold when (a n ) is monotone. 42. If S a n converges, and a n ^ 0, then S B also converges, and also indeed the series 2 /T^TTTa' for ever y ^ > 0. 6 * (051) 126 Chapter IV. Series of arbitrary terms. 43. Every positive real number a^ is, in one and only one way, ex- pressible in the form &$ ^3 ^4 where a n is a non-negative integer with a n S n I for n > 1, subject to the con- dition of not being n 1 for every n after a definite n . If x l is rational, and only then, the series terminates. 44. If <; x < 1 , then there is one and only one sequence of positive integers (A?), with K *i < * 2 < * 3 < - , for which ^ L + _i_ + ... + ___J + ... x is rational if, and only if, the k v 's are all equal after some index v r Chapter IV. Series of arbitrary terms. 15. The second principal criterion and the algebra of convergent series. 00 An infinite scries a n , whose terms are now no longer assumed n=0 subjected to any restriction, but may be arbitrary real numbers, was, we agreed, to be considered as essentially a new symbol for the sequence (s n ) of its partial sums s n = *o + *i H ----- 1- a n (n = , 1 , 2 , . . .) and we proposed to transfer immediately to the series itself the de- signations introduced to characterise the convergence or divergence of (s n ). The case of convergence again occupies our main attention. The second main criterion (47 51), expressing the necessary and sufficient condition for convergence, at once provides the following 81. Fundamental theorem (First form). The necessary and sufficient condition for the convergence of the series 2 a n ts ^at, having chosen any e > , we can assign a number n Q = w (e) such that for every n > n {} and every k ^> 1 , we have that is to say, in the present case, that 15. The second principal criterion and the algebra of convergent scries. 127 Starting with the second form of the main criterion, we also ob- tain for the present fundamental theorem the following Second form. The series 2 a n converges if, and only if, given 81 a, a perfectly arbitrary sequence (k n ) of positive integers, the sequence of numbers T n =(n n+l + ff w+4 H ----- f- r&M+* n ) invariably proves to be a null sequence 1 . And as before we can extend this somewhat to the Third form. The series J a n converges if, and only if, given 81 b. two perfectly arbitrary sequences (vj and (k n ) of positive integers, of % which the first , at least , tends to + oo , the sequence of numbers invariably proves to be a null sequence. R em arks. 1. A series represents essentially a new symbolic expression for se- quences of numbers, and in particular, as we remarked, not only every series represents a sequence, but every sequence is also expressible as a series; all remarks and examples given on p. 84 have their parallels here. 2. The contents of the fundamental theorem may bo formulated as follows: Given e ^> , every portion of the series, however long, provided only its initial index be sufficiently large, must have a sum whose absolute value is <* Or: Given f>> 0, we must be able to assign an index m so that for n^>m the addition, to s n , of an arbitrary number of terms immediately consecutive to a n can only alter this partial sum by less than e. 3. Our present theorems and remarks of course also hold for series of positive terms This the reader should verify in each separate case. A finite part of the series, such as 4- 0K+a H ----- h #*+;i we may for brevity call a pot tion of the series, denoting it by T v if it begins immediately after the y th term. When required, we may further ex- plicitly indicate the number of terms in the portion by denoting this by Ty t i. If we are considering an arbitrary sequence of such portions whose initial index * + > we shall refer to it for short as a "se- qnence of portions" of the given series. The second and third form of the fundamental theorem may then also be expressed thus: 4 th form. The series a n converges if, and only if, every 81 c. "sequence of portions" of the series is a null sequence. 1 It is substantially in this form that N. H. Abel establishes the criterion in his fundamental memoir on the Binomial series (Journ f. die reine u. angew. Math., Vol. 1, p. 311. 1826). 128 Chapter IV. Series of arbitrary terms. , Remarks and examples. 1. Sa n is thus divergent if, and only if at least one sequence of portions can be assigned which is not a null sequence. For the harmonic series , for instance, we have The sequence (T n ) is therefore certainly not a null sequence, and therefore ' is divergent. n 1 2. For y\ 3 we have ** n* 1 . 1 + + ; therefore TV < , so that 7 V ->0, when y-*-f oo. The series therefore converges. 3. For the sequence n=l we have T M = T n . t = (- ] Whether k is even or odd, the expression m brackets is certainly positive and < r . For if we take together, in pairs, each positive term and the follow- n -+ 1 ing negative term, the sum of the two K in each case positive. If k *s even all terms are exhausted in this manner, if k is uneven a positive term remains, so that in either case the complete expression is seen to be positive. If, on the other hand, we write it in the form __ __ ___ - __ n+l \n + 2 w + 3/ Vn + 4 n + 5 all the terms are now exhausted when k is odd and a negative term remains over if k is even, so that in both cases only subtractions from - - occur, n -{- 1 and thus the expression is < - - . As we now have n+ I 1 " ' ' "'"'^n+1 this involves and our series converges. We shall see later (cf. 120) that its sum coincides with the limit of the sequence 46, 2 and has the value log 2. 15. The second principal criterion and the algebra of convergent series. 129 To these four separate forms of the second fundamental criterion we may at once attach the following simple but important considerations: Since in the second form, by putting ft w = l, we obtain a n+1 *0, we have also (by 27, 4), a n *0, i. e. we have the Theorem 1. In a convergent series, the terms a n necessarily form 82, a null sequence: a n -> 0. That this condition is not sufficient for convergence, we know already, from the example of the harmonic series. If, on the other hand, we already know that a n converges, CO then so does the series a M + 1 + n + a + n+3 + =2? a v> whose r-n + l sum is usually, as the so called remainder of the series a n > denoted by r n (so that $ n -\- r n = s = the sum of the complete series). Now we may, in the inequality valid for n > n and every k I> 1 , allow k to increase beyond all bounds and so obtain, for every n > n , r n <^ e . Thus we have the CO Theorem 2\ The remainders r n = a v of a convergent series r=n + l J a n = s , i. e. the numbers n n=o always form a null sequence. In 80, we saw further that if the terms of a convergent series a n (of positive terms) are monotone decreasing, then, over and above the theorem just proved, the condition n a n > must hold. That this need no longer be the case in series of arbitrary terms is already shewn by the series given in 81 c, 3. We can, however, show that we must have <i + 2 gg H ----- h n a n ^ n i. e. that the terms of the sequence (n n ) are small on the average. In fact we have 2 the more general 00 Theorem 3. // a n t5 a convergent series of arbitrary terms n=0 and if (p , p l9 ...) denotes an arbitrary monotone increasing se- quence of positive numbers tending to + oo , then the ratio A a l + + Ai a n ^ Q 2 L. Kronecker, Comptes rendus de 1'Ac. de Paris, Vol. 103, p. 980. 1886. Moreover, this condition is not only necessary, but also, in a quite determinate sense, sufficient, for the convergence of the series 27 a n (cf. Ex. 58 a). 130 Chapter IV. Series of arbitrary terms. Proof. By 44, 2, s n *s implies a l *0 + (P* - &) S l 4- - ' + (ft. -/>-,) *n-i Since--- *0 and s n s, we must therefore have rn W Pn But this is precisely the relation we had to prove, as may be seen at once by reducing to the common denominator p n and grouping in succession the terms which contain P 9p l >->p n respectively 3 . As regards any condition for convergence whatsoever, we have to repeat expressly that the stipulations made therein always concern or only need concern those terms of the series which follow on some determinate one, whose index may moreover be replaced by any larger index. In deciding whether a series is or is not con- vergent, the beginning of the series, as it is usually put for brev- ity, does not come into account. This we express more exactly in the following 00 Theorem 4. // we deduce , from a given series 2 a n > a new 00 series a n ' by omitting a finite number of terms, prefixing a finite n=0 number of terms, or altering a finite number of terms (or doing all three things at once] and now designating afresh the terms of the series so produced by # ', a/, ..., 4 then either both series converge or both diverge. Proof. The hypotheses imply that a definite integer q = Q exists such that from some place onwards, say for every n > m , we have Every portion of the one series is therefore also a portion of the other, provided only its initial index be > m -j- | ? | The fun- damental theorem Sid. immediately proves the correctness of our statement. 8 Instead of the positive p n we may (cf. 44, 3 and 5) take any se- quence (f n ) , for which, on the one hand, | p n \ * -f- oo and, on the other, a constant K is assignable for which for every n. 4 I.e. in short: ". . . by making: a finite number of alterations (27, 4) in the sequence (a n ) of the terms of the scries ..." 15. The second principal criterion and the algebra of convergent series. 131 Remark. It should be expressly noted that for series of arbitrary terms, compari- son tests of every kind become entirely powerless. In particular, of two series 2 a n and 2 a n f whose terms are asymptotically equal (a n ~ a n f ) , the one may f _ ]\n quite well converge and the other diverge. Take for instance a n = -- ft and aJ = a n -\ -- : - . * n nlogn Finally we prove the following criterion of convergence, which appears almost unique in consequence of its particularly elementary character, and relates to the so-called alternating series, i. e. to series whose terms have alternately positive and negative signs: Theorem 5. [Leibniz's rule 5 .] An alternating series, for which the absolute values of the terms form a monotone null sequence, is invariably convergent. The proof proceeds on quite similar lines to that of 81 C, 3. For if a n is the given alternating series, then a n has cither the sign ( l) n , for every n, or the sign ( l) n+1 , for every n. If we write, therefore, | a n \ cc n , we have T n = T n>k = [ B + 1 - n+9 -h cc n+3 -+ + (- I)*- 1 n + J - As the a's are monotone decreasing, we may convince ourselves precisely as in the example referred to, that the value of the square bracket is always positive, but less than its first term a n+1 . Thus \T n \ = [7^1 < w + 1 , which, since cc n forms a null sequence by hypothesis, involves T n * and therefore convergence of a n , by 81 C. The algebra of convergent series. Already in 69, 2, 3, it has been emphasized that the term "sum", to designate the limit of the sequence of partial sums of a series, is misleading in so far as it arouses a belief that an infinite series may be operated on by the same rules as an (actual) sum of a definite number of terms, e. g. of the form (a + b + c + d) > Sa 7- This is not the case, however, and the presumption is therefore fundamentally erroneous, although some of the rules in question do actually remain valid for infinite series. The principal laws in the algebra of (actual) sums are (according to 2, I and III) the associative, distributive and commutative laws. The following theorems arc intended to show how far these laws remain true for infinite series. 6 Letters to /. Hermann of 26. VI. 1705 and to John Bernoulli of 10. 1. 1714. 132 Chapter IV. Series of arbitrary terms. 83. Theorem 1. The associative law holds for convergent infinite series unrestrictedly in the following sense only: a o + a i + *3 H ---- = s implies K + a i -\ ----- h rt n) + fa^+i + 0^+2 -I ----- h -f ---- = s, if T>J , r a , . . . denote any increasing sequence of different integers and the sum of the terms enclosed in each bracket is considered as one term of a new series where, therefore, for k = , 1, 2, ..., ^fe a * k +l + Sc + 2 "I ----- 1" ^'k + i (V Q = 1). The converse is however not always true. Proof. The succession of partial sums S^ of 2 A k is ob- viously the sub-sequence s ri , s r , f ..., s r , ... of the sequence of partial sums s n of 2a n . By 41, 4, 5 M therefore tends to the same limit as s n . Remarks and examples. 00 / _ J\ 1 111 1. The convergence of y\ - - ' - ==1 --f- % - --- - --- . therefore im- n=l n ^ J 4 plies that of 00/1 IN 00 1 111 vi I * ____ L j _ y _ * ___ _ _^ __ - ___ A __ ^ tSi \2 A - 1 2 A/ "" ^ (2 A - l)-2 A "" 1 -2 "^ 3-4 "^ 5-6 "*" " " * and also, similarly, of i (^-M-f 1 M-...-I -L- JL. x .. '"VT T/ U~T/ ==1 "2.3 4-5 (T? and all three series have the same sum. If we denote this by s, the second 117 series shows that in any case, s > r~ o + o~~T = To and tlie tnird > tnat 1 u O*4 \u 12 ^"^12' 2. That we may introduce brackets, but may not without consideration omit brackets occurring m a series, the following simple example shows: The series + + + . . . is certainly convergent and has the sum 0. If we substitute every- where (1 1) for 0, we obtain the correct equality (1 - 1) + (1 - 1) + . . . = 2(1 - 1) - 0. But by omitting the brackets we obtain the divergent series 1_ 1 + 1 _ 1 + _..., 15. The second principal criterion and the algebra of convergent series. 133 which therefore may not be put "= 0". For we should then by again grouping the terms, though in a slightly different way, obtain 1 (1 1) (1 - 1) ... s 1 _ - - . . . , which again converges and has the sum 1. We should therefore finally deduce that = 1 ! ! e . We proceed at once to complete Theorem 1 by the following CO Theorem 2. If the terms of a convergent infinite series 2 A k are *=o themselves actual sums (say, as above, A k a^ k+1 + . . . + ^ fc+1 ; 0, !>; ^o 1)> tnen we "may" omit the brackets enclosing these if % 00 and only if, the new series 2! a n thus obtained also converges. n=0 In fact in that case, by the preceding theorem, Z a n 2 A k , while in the case of divergence of 27 a n , this equality would become meaningless. A usually sufficient indication as to whether the new series converges is provided by the following Supplementary theorem. The new series Ea n deduced from H A k in accordance with the preceding theorem is certainly convergent if the quantities form a null sequence 7 . Proof. If s be given > 0, choose m^ so large that, for every k > m l9 we have and choose m% so large that, for every k > m 2 , we have A k ' < ?. If m 2 is larger than both these numbers m l and m z , then we have, for every 8 In former times before the strict foundation of the algebra of infinite series (v. Introduction) mathematicians found themselves fairly at a loss when confronted with paradoxes such as this. And even though the better mathematicians instinctively avoided arguments such as the above, the lesser brains had all the more opportunity of indulging in the boldest speculations. Thus e. g. Guido Grandi (according to R. Reiff, v. 69, 8) believed that in the above erroneous train of argument which turns into 1, he had obtained a mathematical proof of the possibility of the creation of the world from nothing ! 7 As A k - 0, this is of itself the case if the terms which constitute A k have one and the same sign in particular, therefore, if by omission of the brackets we obtain a series of positive terms. Furthermore, this is always the case if the terms a n form a null sequence and if the number v k +i v k of terms grouped together in A k forms a bounded sequence for A 0, 1, 2, . . . (An example is afforded by the series 2(a n -f- b n ) in the next Theorem 3.) 134 Chapter IV. Series of arbitrary terms. For to each such n corresponds a perfectly definite number A, for which and this number k must be ^> m. In that case, however, 5 n = 5 *-l + , fc + l H ----- I- *n> K - Sfc-i I ^ 4 fc ' < And since S n ~ S = ( 5 n ~ 5 fc-l) + ( 5 /c-l - 5 ) we then have, effectually, a n = s > q- e.d. n=0 Example. is convergent; for ^ is positive, and, for every fc> 1, is ..2 1 1 1 ^4^-4 2^5 2(*-l) A* 1 **" Since similarly, for every k > 1 , 211 ' ') is a null sequence. Therefore the series is also convergent. Its sum call it S is certainly >> A^ -f- A 2 > ~- , as 1 ^ the series in its first form had only positive terms. Theorem 3. Convergent series may be added term by term. More precisely, JX = s and 2* n = t n=-o w=o imply both l(. + j-+ n=o and also without brackets I a o + 6 o + *i + & i + a H ---- = + * 15. The second principal criterion and the algebra of convergent series. 135 Proof. If s n and t n are the partial sums oj the first two series, then (s n + O are those of the third. By 41, 9, it therefore follows at once that (s n -f- n ) - * s + t. That the brackets may be omitted, in the series thereby proved convergent, follows from the supplementary theorem of Theorem 2, since (|0 n |) and (|& n |) and therefore also (| a n \ + | b n |) are null sequences. Theorem 4. Convergent series may in the same sense be sub- tracted term by term. The proof is identical. Theorem 5. Convergent series may be multiplied by a constant, that is to say, from 2a n = s it follows, if c is an arbitrary number , that Proof. The partial sums of the new series are cs n , if those of the old are s n . Theorem 41, 10 at once proves the statement. This theorem, to some extent, provides the extension to infinite series of the distributive law. Remarks and Examples. 84. 1. These simple theorems are all the more important, as they not only allow us to deduce the convergence of the new series from the convergence of known series, but also set up a relation between its sum and that of the known series. They form therefore the foundation for actual calculation in terms of infinite series. QO / _ J)n -1 2. The series - - - - was convergent. Let s denote its sum. By n=l n theorem 1, the series J __ M /i-l 2ft/ and -l 4k are then also convergent with the sum s. Multiply the first by , in accor- dance with Theorem 5, this giving -2 4 kJ 2 and add this term by term to the second; we obtain 3 1 JL] - 1 2 ft/ 4 k - 3 ^ 4 k n> a. or more precisely: we obtain the convergence of the series on the left hand side and the value ot its sum, the latter expressed in terms of the sum of the series from which we started. The convergence was also proved directly in connection with theorem 2; the present considerations have led however appre- ciably further, since they afford a definite statement as to the sum of the series. Before we examine the validity of the commutative and distributive laws and investigate, in relation to the latter, the possibility of forming the product of two series, we still require an important preliminary. 136 Chapter IV. Series ol arbitrary terms. 16. Absolute convergence. Derangement of series. The series 1 -3 + | 3- + proved (81 c, 3) to be convergent* But if we replace each term by its absolute value, the series becomes the divergent harmonic series 1 + | + | + In all that follows, it will usually make a very material difference whether a convergent series 2 a n remains convergent or becomes divergent, when all its terms are replaced by their absolute values. Here we have, to begin with, the 85. o Theorem. A series 2a n is certainly convergent if the series (of positive terms') 2\ a n \ converges 8 . And if Za n = s, 2\ a n \ = 5 then l|^s. Proof. Since the left hand side is here certainly < e if the right hand side is, whence by the fundamental theorem 81 our first statement at once follows. Since further we have also, by 41, 2, | s | < S. By this theorem, all convergent series are divided into two classes and 2a n belongs to the one or the other according as -2"|fl,J is or is not also convergent. We define 86. Definition. If a convergent series Ea n is such that Z\ a n \ also converges, then the first series will be called absolutely convergent, and other- wise non-absolutely convergent 9 . Examples. The series ( ^ ; 5l^t2?. (>!); -l n w-1 n =0 are absolutely convergent. Every convergent series of positive terms is of course absolutely convergent. The very great significance of the concept of absolute convergence will first appear in this : the convergence of absolutely convergent series is much more easy to recognise than that of non-absolutely convergent series, usually, in fact, by comparison with series of positive terms, 8 Cauchy, Analyse alggbrique, p. 142. (The proof is inadequate.) On the other hand, the example just given showed that the convergence of 2a n need not involve that of 2\a n \. 9 A series is thus u non-absolutely convergent" if it converges, but not absolutely. The designation "non-absolutely convergent" applies therefore to convergent series only. 16. Absolute convergence. Derangement of series. 137 so that the simple and far-reaching theorems of the preceding chapter become available for the purpose. But this significance will imme- diately become further visible in that we may operate on absolutely convergent series, on the whole, precisely as we operate on (actual) sums of a definite number of terms, whereas in the case of non-absolutely convergent series this is in general no longer the case. The following theorems will show this in detail. Theorem 1. // 2c n is a convergent series of positive terms and 87. if the terms of -a given series 2 ' a n , for every n > m, satisfy the condition I a n I ^ c n or ^0 condition then 2a n is (absolutely) convergent. Proof. By the 1 st and 2 nd comparison tests, 72 and 73, respec- tively, | a n | is in either case convergent 11 , and so therefore, by 85, is 2 a n . In consequence of this simple theorem the complete store of con- vergence tests relating to series of positive terms becomes available for series of arbitrary terms. We infer at once from it the following Theorem 2. // Za n is an absolutely convergent series and if the factors a n form a bounded sequence, then the series is also (absolutely) convergent. Proof. Since (|# n |) is a bounded sequence simultaneously with (aj, it follows from 70,2 that 2*| a n |-| a n | = 2*| # n a n | is convergent simultaneously with 2 \ a n \ . Examples. 1. If 2 c n is any convergent series of positive terms and if the c^, 's are bounded, then 2 ct n c n is also convergent, for then 2 c n is also absolutely con- vergent. We may thus, for instance, instead of joining the terms c t *!>*> with the invariable sign +, replace this by quite arbitrary + .and signs, in every case we get a convergent series; for the factors ^ 1 certainly form a bounded sequence. Thus for instance the series a , S(-V)c a are all convergent, where [z], as usual, stands for the largest integer not greater than z. 10 In the second condition, it is tacitly assumed that, for every n>m, a n 4= and c n =(= 0. 11 The corresponding criteria of divergence, dm and are of course abolished, since the divergence of J?| a n |, not necessarily of is all that follows. Cf. Footnote 8. 138 Chapter IV. Series of arbitrary terms. 2. If Sa n is absolutely convergent, then the series obtained from it by an arbitrary alteration in the signs of its terms, is invariably an absolutely convergent series. We shall now returning thereby to the questions put aside at the end of last section ( 15), show that for absolutely convergent series the fundamental laws of the algebra of (actual) sums are in all essen- tials maintained, but that for non- absolutely convergent series this is no longer the case. Thus the commutative law "a-\-b = 6 -f- a" does not in general hold for infinite series. The meaning of this statement is as follows: If (v Q> v iJ v a , . . .) is any rearrangement (27,3) of the sequence (0,1, 2, . . .) then the series !;<*' = j>j a v (i. e. with a n ' = a v for n = 0, 1, 2, . . .) n =0 n=0 w n 00 will be said, for brevity, to result from the given series a n by =o rearrangement or derangement of the latter. The value of (actual) sums ot a definite number of terms remains unaltered, however the terms may be rearranged (permuted). For infinite series this is no longer the case 1 *. This is shown already by the two series considered as examples in 81 c, 3 and 83 Theorems 1 and 2, namely which are evidently rearrangements of one another, but have different sums. The sum of the first was in fact s < j, while that of the second was s'>^; and indeed the considerations of 84, 2 showed more precisely that s' = f s. This circumstance of course enforces the greatest care in working with infinite series, since we must to put it shortly take account of the oider of the terms 18 . It is therefore all the more valuable to know in which cases we may not need to be so careful, and for this we have the 88. Theorem 1. For absolutely convergent series , the commutative law holds unrestrictedly 1 *. Proof. Let Z a n be any absolutely convergent series (i. e. Z \ a n \ is convergent as well), and let Z a n ' =27 a v be a derangement of Z a n . M Thib was first remarked by Cauchy (Resume's analytiques, Turin 1833). 18 As 2 a n merely represents the sequence (s n ) t and a rearrangement of Za n produces a series 2 a n ' with entirely different partial sums s n ', these not merely forming a rearrangement of (s n ), but representing entirely diffe- rent numbers!! it seems a priori most improbable that such a derangement will be without effect on the behaviour of the series. 14 Lejeune-Dinchlet, G.: Abh. Akad Berlin 1837, p. 48 (Werke I, p. 319). Here we also find the example given in the text, of the alteration in the sum of the series by derangement. 16. Absolute convergence. Derangement of series. 139 Then every bound for the partial sums of S \ a n \ is clearly also a bound for the partial sums of 27 1 a n ' |. So Z a n ' is absolutely convergent with S a n . Let s n denote the partial sums of 2 a n , and s n ' those of S a n '. Then if e is arbitrarily given > 0, we may first choose m, in accordance with 81, so large, that for every k ^ 1 I + I m+ 2 1 + + | a m+k I < e and now choose n Q so large that the numbers v , v l9 v 2 , . . . , v n comprise 15 at least all the numbers 0, 1, 2, . . . , m. Then the terms a , a lf a z , > a m evidently cancel in the difference s n ' s n , for every n > w , and only terms of index > m remain, that is, only (a finite number of the) terms a m\-i> a m\2> Since, however, the sum of the absolute values of any number of these terms is always < e, we have, for every n > n Q , and therefore (s n r s n ) is a ' null sequence. But this implies that s n = s n + ( s n ~ s n) nas tne same limit as s n , i. e. a n ' is convergent and has the same sum as 2 a n , q. c. d. This property of absolutely convergent series is so essential that it deserves a special designation: Definition. A convergent infinite series which obeys the commutation 89. law without any restriction , i. e. remains convergent, with unaltered sum, under every rearrangement, shall be called unconditionally conver- gent. A convergent series, on the other hand, whose behaviour as to con- vergence can be altered by rearrangement, for which therefore the order of the terms must be taken into account, shall be called conditionally con- vergent. The theorem proved just above can now be expressed as follows: ''Every absolutely convergent series is unconditionally convergent" The converse of this theorem also holds, namely Theorem 2. Every non-absolutely convergent series is only condi- tionally convergent 16 . In other words, the validity of the equality 00 2 = * H~0 in the case of a non-absolutely convergent series Z a n depends essentially on the order of the terms of the series on the left, and may therefore, by a suitable rearrangement, be disturbed. 16 That such a number n Q exists follows from the very definition of derange- ment. 16 Cf. Fundamental theorem of 44. 140 Chapter IV. Series of arbitrary terms. Proof. It obviously suffices to prove that, by a suitable rearrange- ment, we can deduce from 2 a n a divergent series 2 a n '. This we may do as follows: The terms of the series 2 a n which are ^ 0, we denote, in the order in which they occur in 2 a n , by p l9 p 2> p& . . . ; those which are < we denote similarly by q l9 <? 2 , q 3 , . . . Then 2 p n and 2 q n are series of positive terms. Of these, one at least must diverge. For if both were convergent, with sums P and Q say, then we should obviously have, for each n, hence 2 a n would, by 70, be absolutely convergent, in contradiction with our assumption 17 . If for instance 2 p n diverges, then we consider a series of the form Pi +P2 + + Am 01 + Ai+l + Am+2 + + Pm* ft + A.+l + > in which, therefore, we have alternately a group of positive terms fol- lowed by a single negative term. This series is clearly a rearrangement of the given series 2 a n and will, as such, be denoted by 2 a n '. Now since the series 2p n was assumed to diverge, and its partial sums are therefore unbounded, we can, in the above, first choose m so large that p -f p 2 + + Pm L >l + ?i> tncn m z > m i so large that Pi+P* "!- + Am + - +p m > > 2 + ?i + ft and, generally, w,, > #*_! so large that + Pm v > v + q l + ft + . . . + q (y = 3, 4, . . .). But 2 a n ' is then clearly divergent; for each of those partial sums of this series whose last term is a negative term q v of 2 a n , is by the above > v (v = 1, 2, . . .). And since v may stand for every positive integer, the partial sums of 2 a n ' are certainly not bounded, and 2 a n ' itself is divergent, q. e. d. 18 . If 2 q n is divergent, we need only interchange 2p n and 2 q n suitably in the above to reach the same conclusion. 17 It is not difficult to see that actually both the series Sp n and q n must diverge (cf. 44); but this is for the moment superfluous. 18 E a n ' clearly diverges to + 00. 16. Absolute convergence. Derangement of series. 141 27 > - '- E= _i---__--- _ 71=1 n * & < convergent. Since (cf. 46, 3), for A =- 1, 2, . . . , Example. ... was seep to De non-absolutely we have, for v 1, 2, . . . , !+l + J+---+2; >2v - If therefore we apply to the series 2 - the procedure described above, we ti need only put m v ~ 2 8 ", to deduce from it by rearrangement the divergent series 2 4- 4 + ^ + ... + 28 ~ 1-f 2 8~4r~2 ~*~ ' ' ' + 2 ~~ 3 "*" ' * ' For the partial sums of this series terminating with the v th negative term is greater than 2 v minus v proper fractions, i. e. certainly > v. Theorem 88, 1 on the derangement of absolutely convergent series may still be considerably extended. For the purpose, we first prove the following simple Theorem 3. If H a n is absolutely convergent, then every "sub-series" Sa\ n for which the indices A n denote, therefore, any monotone increasing sequence of different positive integers, is again convergent and in fact again absolutely convergent. Proof. By 74, 4, 27 1 a Xfl \ converges with E (a n ). By 85, the state- ment at once follows. We may now extend the rearrangement theorem 88, 1 in the fol- lowing manner. We begin by picking out a first sub-series 27 a\ n of the given absolutely convergent series 27 a n , and arranging this first sub-series in any order, denote it by let # (0 > be the sum of this series, certainly existing, by the preceding theorem, and independent of the chosen arrangement by 88, 1 19 . We may also allow this and the following sub-series to consist of only a finite number of terms, i. e. not to be an infinite series at all. From the remaining 19 The letter z is intended as a reference to the rows of the following doubly infinite array. 142 Chapter IV. Series of arbitrary terms. terms as far as is possible we again pick out a (finite or infinite) sub-series, and denote it, arranged in any order, by its sum by # (1) ; from the remaining terms we again pick out a sub-series, and so on. In this manner, we obtain, in general, an infinite series of finite or (absolutely) convergent infinite series: If the process was such as to give each non-zero term 20 of the scries a n a place in one (and only one) of these sub-series, then the series or, that is to say, the series may in a further extended sense be called a rearrangement of the given series 21 . For this again we have, corresponding to theorem 88, 1 : Theorem 4. An absolutely convergent series "may" also in the ex- tended sense be rearranged. More precisely: The series * + *d> + *< 2) + is again (absolutely) convergent, and its sum is equal to that of a n . Proof. If e > be given, first determine m so that, for every k ^ 1 , the remainder | a m+1 \ + | a m+2 \ + . . . < e, and then choose n so that in the first n Q + I sub-series 2 a n ^\ v = 0, 1, . . . , # , the terms a Q , a ly a 2 , . . . , a m of the given series certainly appear. If n > n Q and > m, then the series . . . + *0> - s n 20 The introduction or omission of zero terms in 27 a n or in the partial sums is obviously without influence on the present considerations. 21 Put into the first sub-series, besides a and a jt all those terms a nt for in- stance, in any order, whose indices n are divisible by 2; into the next all those of the remaining terms whose indices are divisible by 3; into the next again all re- maining terms whose indices are divisible by 6; and so on, using the prime numbers 7, 11, 13 ... as divisors. 16. Absolute convergence. Derangement of series. 143 contains only terms a n whose indices are > m. Hence, by the choice of m, the absolute value of this difference is < e, and tends therefore, with increasing , to zero, so that lim (*(> + *W + . . . + * (n) ) = lim s n = s = 27 a n . n >*) Moreover, the convergence of 27 #W which is thus established is also absolute, since for each n we have obviously The converse of this theorem is, of course, even less valid than that of theorem 83, 1, without further consideration. Given, for k - 0, 1 , 2, . . . , the convergent series *<*> = 27* n <*> f if the aggregate of terms a n W be arranged in any way as a sequence (cf. OC 53, 4), then 27 a n need not at all converge, even should 27 zW be con- k-o vergent. To show this is possible we have only to take, for each of the series z^ k \ the series 1 1 -|- + + + - . And even if 2 a n con- verges, the sum need not be equal to that of 2 zW. A general discussion of the question under what circumstances this converse of our theorem does hold, belongs to the theory of double series. However, we may even here prove the following case, which is a par- ticularly important one for applications : Main rearrangement theorem 22 . We suppose given an infinite 90. number of convergent series = a (o) + fll <o) + . . . + a n <> + . . . *0> = *bO> + *!<'>+... + ,<!>+... (A) <*> and assume that these series are not only absolutely convergent, but satisfy the stricter condition that, if we write 27| *<*> | = W (* = 0, 1, 2, . . . , fixed), =o the series 22 Also called Cauchy's Double Series Theorem. 144: Chapter IV. Series of arbitrary terms. is convergent. Then the terms standing vertically one below the other also form (absolutely) convergent series; and if we write 23 = & ( = 0, 1,2,... .fixed), =o then ZsW is again absolutely convergent and we have = J7*<*>; in other words, the two series formed by the sums of the rows and by the sums of the columns, respectively, are both absolutely convergent and have the same sum. The proof is extremely simple: Suppose all the terms in (A) arranged anyhow (in accordance with 53, 4) in a simple sequence, and denoted, as terms of this sequence, by a , a l9 a 2 , . . . . Then H a n is absolutely con- vergent. For every partial sum of 27 \ a n \ , for instance must still be ^ o-, since by choosing k so large that the terms a , a l9 a 2 , . . . , a m all occur in the k first rows of (A), we certainly have i. e. fg a. A different arrangement of the terms a n W in (A) as a simple sequence a ', a^, a%, . . . would produce a series 2 a n ' which would be a mere rearrangement of 2 a n , and therefore again absolutely convergent, with the same sum. Let this invariable sum be denoted by s. Now both 2zW and also 2sW are rearrangements of a n = s, in the extended sense of theorem 4, just proved. Therefore these two series are both absolutely convergent and have the same sum s, q. e. d. This rearrangement theorem may be expressed in somewhat more general form as follows: Supplementary theorem. If M is a countable set of numbers and there exists a constant K such that the sum of the absolute values of any finite number of the elements of M remains invariably < K, 23 Here the letter s is intended as a reference to the columns of (A). 16. Absolute convergence. Derangement of series. 145 then we can assert the absolute convergence with the invariable sum s of every series 2 A k whose terms A k represent sums of a finite or infinite number of elements of M (provided each element of M occurs in one and only one of the terms A 1( ). And this remains true if we allow a repetition of the elements of M, provided each ele- ment occurs exactly the same number of times in all the A k 's taken together, as in M itself**. Examples of these important theorems will occur at several crucial points in what follows. Here we may give one or two obvious applications: 1. Let 2a n =s be an absolutely convergent series and put t _+4a_ a + ... + 2 "a M = a/ (w = 0, 1, 2, . . .) Then we also have 2a n ' = s. The proof results immediately, by the previous rearrangement theorem, from the consideration of the array aa= 0+ +4-+4 2. Similarly, from the array 1-2" 7 " 2.3" 1 " 3-4" + 2 2^ + V4- >= o + 8^- -.~ (v. 68, 2h), and we deduce the equality, valid for any absolutely convergent series 2-3 a o + 2 3 4 8. The preceding rearrangement theorem evidently holds whenever every ***** is S ^ and at least one ^ tlie two series 2zW and ^s("> converges; it holds further whenever it is possible to construct a second array (A') similar to (A), whose terms are positive and > the absolute values of the corresponding terms in (A), and such that, in (A'), either the sums of the rows or the sums of the columns form convergent series. 24 An infinite number of repetitions of a term different from zero is ex- cluded from the outset, since otherwise the constant K of the theorem would certainly not exist. And the number can produce no disturbance. 146 Chapter IV. Series of arbitrary terms. 17. Multiplication of infinite series. We finally enquire to what extent the distributive law "a (b -f- c) = a b + c" holds for infinite series. That a convergent infinite series 2a n may be multiplied term by term by a constant, we have already seen in 83, 5. In the simplest form the distributive law is therefore valid for all convergent series. In the case of actual sums, it at once follows further, from the distributive law, that (a-\-fy(c-\-d)=^ac-\-ad -\-bc-\-bd, and more generally, that (a + a l +.-- + or in short, that /i=0 ^<=o, . ..,m where the notation on the right is intended to convey that the indices A and ju, assume, independently of one another, all the integral values from to I and to m respectively, and that all (I -f- 1) (m -f- 1) such products a^ b^ are to be added, in any order we please. Does this result continue to hold for infinite series? If 2a n ~s and 2b n = t are two given convergent infinite series of sum s and t> is it possible to multiply out in the product (00 v / GO \ 2*i}-( 2W A=0 / \/,=0 / in any similar way, and in what sense is this possible? More precisely: Let the products a* &/i be denoted, in any order we choose 25 , by , p^ p 29 . . . ; is the series 2p n convergent, and if convergent, does it have the sum s t ? Here again absolutely convergent series behave like actual sums. In fact we have the Theorem 26 . If the series Za n = s and Zb n = t are absolutely convergent^ then the series Sp n also converges absolutely and has the sum s t. 25 We suppose, for this, that the products a\ b^ are written down exactly in the same way as n (t) or aM for 53, 4 and 90, to form a doubly infinite array (A). We can then suppose in particular the arrangement by diagonals or the arrangement by squares carried out for these products. 26 Cauchy: Analyse alglbrique, p. 147. 17. Multiplication of infinite series. 147 Proof. 1. Let n be a definite integer > and let m be the largest of the indices i and JLL of the products a\ b^ which have been denoted by p Q , p , ..., p n - Evidently i.e. < a r, if (7 and T denote the sums of the series 2, \ a^ \ and J? | & ; < | . The partial sums of Jfj^ n | are therefore bounded and 2p n is ab- solutely convergent. 2. The absolute convergence of 2 p n having been proved, we need only determine its sum call it S for a special arrangement of the products a^ b^, for instance the arrangement "by squares". For this we have, however, obviously, o b o = Po> K + i)(*o + 6 i) = Po + Pi + P* + PB and in general an equality which, by 41,10 and 4, becomes, when n *>oo, which was the relation to be proved. Remarks and Examples. 1. As remarked, for the validity of the relation 2 p n st under the hypo- theses made, it is perfectly indifferent in what manner the products a a b are " " enumerated, that is to say arranged in order as a simple sequence (p n ). The arrangement by diagonals is particularly important in applications, and leads, if the products in each diagonal are grouped together (83, 1), to the following relation : n=0 n=o writing for brevity a b n + &i &_! H- 2 & n-a + + <* n *> = c n . The validity of this relation is therefore secured when both series on the left converge ab- solutely. We are also led to this form or arrangement of the ''product series 11 , sometimes called Catichy's product of the two given series 27 , by the conside- ration of products of rational integral functions and those of power series, which latter will be discussed in the following chapter: If in fact, we form the pro- duct of two rational integral functions (polynomials) and b -f- 6 X x H- b v x 2 -\ ----- h b m x m Cauchy loc. cit. examines the product series in this special form only. 148 Chapter IV. Series of arbitrary terms. and arrange the result again in order of increasing powers of x, then the first terms are o b o + ( a o *>i + a i 5 o) * + ( & + a i b i + a &o) * a H ---- so that we have the numbers c , c lt c 2 , - , above introduced, appearing as coefficients. It is precisely due to this connection that Cauchy*s product of two series occurs particularly often. 2. Since 2x n is convergent for |#|<1, we have for such an x n=0 n==0 M iC n 3. The series J? r, cf. 76, 5c and 85, is absolutely convergent for nl every real number x. If therefore x^ and a; a are any two real numbers, we may form the product of according to Cauchy's rule. We get Therefore we have for arbitrary x t and a; a putting o^ + a? f = ar f : v fL. v f = ffl n-O W ^ " n -0 M! n = w! * By our theorem, we have now established that the distributive law may at any rate be extended without change to infinite series, and this, moreover, with an arbitrary arrangement of the products a A bp , if both the two given series are absolutely convergent. It is conceivable that this restricting assumption is unnecessarily strict. On the other hand, the following example, given already by Cauchy 2 * for the purpose, shows that some restriction is necessary, or the theorem no longer holds: Let so that 2a n and 2b n are convergent in accordance with Leibnitz's rule 82, 5. Then c = c == 0, and for n ^> 2, + Replacing each root in the denominators by the largest, Vn 1 , it follows that, for n ^ 2, and therefore the product series 2c n ~ 2 (a Q b n + a 1 6 W-1 + + 18 Analyse alg^brique, p. 149. Exercises on Chapter IV. 149 is certainly divergent in accordance with 82, 1. This is therefore a fortiori the case when we omit the brackets. Nevertheless, the question remains open, whether we may not be able, under less stringent conditions than that of absolute conver- gence of both the series 2 a n and 2 b n , to prove the convergence of the product series 2 p n at least for some special arrangement of the terms a^b^y for instance as in the series 2 c n above. To this question we shall return in 45. Exercises on Chapter IV, 45. Examine the convergence or divergence of the series 2(- -l) n a n , for which a n , from some n onwards, has one of the following values: 1111 1 11 , (-1)* r " H * an + b' J n ' log n 1 log log ' n > i n y n r 46. What alterations have to be made in the answers to Ex. 34, when the behaviour of ( l) n a n is required? 47. Let for 2** <n<2 2 *- fl , ^ (* = 0, 1, 2, ...) i - - ~Q Z..LI ^ -XV'>Z--L$> ** iii/ [1 for Then the series 00 f V K~2 nlogn Converges. What is the behaviour of ^ ~ ? ^^ n co 2 n 4- 1 48. ( I)""" 1 - / , j\ is convergent and has the sum 1. 49. Let the partial sums of the series 1 ^- -f- -\ ... be & o 4 denoted by s n , and its sum by s, and put -| -f- -f- ^ = x n . Show that, for every n, oo / J\n-l so that lim x n = = s (= log 2). 50. Let s(= log 2) denote as above the sum of the series 1 o" + "o h Prove the following relations: x 1 1 1 , * 1 ! , l l l , 1 x i o a > ^s-^s-S-ii+v-Ig-u 4 " -g""5 10 8; W 1 l 1 4 . 1 . 1 -1 + 1 - 1 - ! 4. l *> 1 ~2 4^3 6 8^5 10 12 + ""2 v , 1 1 1 , 1 * 1 , , 2 i o C ) 1 -2""4" h 5 + 7""8~10 + + = o log 2; <5 4 O / O L\J O d) 1+ j + l J_J + + + 1,0*6; o o z 4 J e) I-4 + J-!- J + + + , o u 6 4 o 150 Chapter IV. Series of arbitrary terms. 51. With reference to the last two questions, show generally that the series remains convergent when we alternately write throughout p positive terms 1 P and q negative ones, and that the sum is then = log 2 + ~ log. J q 52. The harmonic series l-f.---f--{- T -{- ... remains divergent, when the i O 4 signs are so changed that we have throughout alternately p positive terms and q negative ones, with p ^ q . If p = q the resulting series is convergent. oo / iyi-i 53. Consider the rearrangements of the series ^ == exactly corre- n=l \/ w ( l) w ~* sponding to those of the series 5] in Ex. 50 and 61. When is the resulting series convergent and when is it not? When is the sum expressible in terms of the sum of the given series? 54. Consider, with the series J^pr, the same alterations in signs as in v^ Ex. 52, for the series JJJ . When is a convergent series obtained? 55. For which values of tx do the following two series converge: 1 _J. + I_l+i_+-..., 2 3 4 5 6" 1+ .L_! + lH. !_ 3 a 2 a 5 a 7 a 4 56. The sum of the series 1 1 1 lies between -- and 1, 2 a 3 a 4" <* for every a > 0. 57. Given show that , J_, 1 , 1 +-I+. .-5s T 5 a f ^ a " t- ip -r 13 a~ T "" 3 ' _ , , __ _-4.J. ~ 2 a 4 a ~ + 5 2+ 7* 8 a lO 2 "^^" "*"~9 S ' (With the latter equality cf. Ex. 50 c.) 58. Tn every (conditionally) convergent series the terms can be grouped together in such a manner that the new series converges absolutely. 58 a. The following complement to KroneckeSs theorem 82, 3 holds good: If a series 2 a n is so constituted that for every positive monotone sequence (/>) tending to -f oo , the quotients Pn tend to 0, then S a n is convergent. In this sense, therefore, Kronecker's condition is necessary and sufficient for the convergence. 18. The radius of convergence. 151 59. If from a given series 2a n , with the partial sums s n> we deduce, by association of terms, a new series 2 A k with the partial sums S&, then the inequalities lim s n ^ lirn S^ ^ lirn S k ^ lim s n invariably hold good, whether 2 a n converges or not. 60. If Sa m , with the partial sums s n , diverges indefinitely, and s' is a value of accumulation (5) of the sequence (s n ), then we can always deduce from 2a A , by association of terms, a series 2 A^ converging to s' as sum. 61. If ~0 n with the partial sums s n , diverges indefinitely, and -*(), then every point of the stretch between the upper and lower limits of s n is a point of accumulation of this sequence. 62* If every sub-series of 2 a n converges, then the series itself is absolutely convergent 63. Cauchy's product of the two definitely divergent series /8V /3V a- and that of the two series 3 + 27 3 n and - 2 + 27 2 W is 6-fO-f-O + O-f-.... -l n-l In both cases it is absolutely convergent. How can this paradox be explained? Chapter V. Power series. 18. The radius of convergence. The terms of the series which we have examined so far were, for the most part, determinate numbers. In such cases the series may be more particularly characterised as having constant terms. This however was not everywhere the case. In the geometric series a n , for instance, the terms only become determinate when the value of a is assigned. Our investigation of the behaviour of this series did not, consequently, terminate with a mere statement of convergence or divergence, the result was: 2a n converges if\a\ < 1, but diverges t/|#|^>l. The solution of the question of convergence or divergence thus depends, as do the terms of the series themselves, on the value of a quantity left undetermined a variable. Series which have their terms, and accordingly their convergence or divergence, depending on a variable quantity (such a quantity will usually be denoted by x 152 Chapter V. Power scries. and we shall speak of series of variable terms 1 ) will be investigated later in more detail. For the moment we propose only to consider series of the above type whose generic term, instead of being a number a n , has the form *" i. e. we shall consider series of the form 2 + a,x + a,x* + ... + *" + ... ^ *". n^O Such series are called power series (in x), and the numbers a n are their coefficients. For such power series, we arc thus not concerned simply with the alternatives "convergent" or "divergent", but with the more precise question: For what values of x is the series convergent, and for what values divergent? 92. Simple examples have already come before us: 1. The geometric series 2 x n is convergent for |as|<l, divergent for | x | ^ 1 . For | x | <1 1, indeed, we have absolute convergence. x n 2. J? - is (absolutely) convergent for every real x; likewise the series 8. JjjJ , because < | a; | n , is absolutely convergent for | a: [ < 1 . For | x | > 1 , the series is divergent, because in that case (by 88, 1 and 40), x n . _j_ oo . For x = 1 it reduces to the divergent harmonic series, and for n x = 1, to a series convergent by 82, Theorem 5. oo y.n 4. ~~2~on * s (absolutely) convergent for ||^2, but divergent for 5. X l* n a; n is convergent for # = 0; but for n;^ value of x ^-- it is n=l divergent, for if x ={= 0, | x \ -+> -f oo and a fortiori \ n* x n \ -* -foo , so that (by 82, Theorem 1) there can be no question of the series converging. For x = 0, obviously every power series 2a n x n is convergent, whatever be the values of the coefficients a n . The general case is evidently that in which the power series converges for some values of x, and diverges for others, while, in special instances, the two extreme cases may occur, in which the series converges for every x (Example 2), or for none =|= (Example 5). 1 The harmonic series ^ ^ is also of this type: it converges for ?> 1, diverges for x < 1. * We here write, for convenience, x = 1, even when x 0. 18. The radius of convergence. 153 In the first of these special cases we say that the power series is everywhere convergent, in the second leaving out of account the self-evident point of convergence x = we say that it is nowhere convergent. In general, the totality of points x for which the given series 2a n x n converges is called its region of convergence. In 2. this consists therefore of the whole axis of x, in 5. of the single point 0; in the other examples, it consists of a stretch bisected at the origin, sometimes with, sometimes without one or both of its endpoints. In this we may see already the behaviour of the series in the most general case, for we have the Fundamental theorem. If 2a n x n is any power series which 93. does not merely converge everywhere or nowhere, then a definite positive number r exists such that 2a n x n converges for every \ x \ < r (indeed absolutely), but diverges for every \ x \ > r. The number r is called the radius of convergence, or for short the radius, and the stretch r . . . + r the interval of convergence, of the given power series 3 . Fig. 2 schematizes the typical situation established by this theorem. dw -r U + r dor. Fig 2. The proof is based on the following two theorems. Theorem 1. // a given power series 2a n x n converges for x = # (X Q 4= 0), or even if the sequence (a n x n ) of its terms is only bounded there, then 2 a n x n is absolutely convergent for every x = x l nearer to the origin than x Q9 i. e. with \x { \ < \X Q \. Proof. If | a n x Q n \ < K, say, then where # = the proper fraction . By. 87, 1 the result stated follows x o immediately. * Theorem 2. // the given power series 2 a n x n diverges for x = x then it diverges a fortiori for every x = x further from the origin than x , i. e. with | * | > | * 1 3 Jn the two extreme cases we may also say that the radius of conver- gence of the series is r = or r:=-f-oo, respectively. 154 Chapter V. Power series. Proof. If the series were convergent for x lt then by theorem 1 it would have to converge for the point a? , nearer than aJ A , which contradicts the hypothesis. Proof of the fundamental theorem. By hypothesis, there exists at least one point of divergence, and one point of convergence 4= 0. We can therefore choose a positive number X Q nearer than the point of convergence and a positive number y further from than the point of divergence. By theorems 1 and 2, the series 2 a n x n is convergent for x = x Q , divergent for ce = ;y , and therefore we certainly have X Q < y . To the interval 7o = ^o !Xo> we a PPty ^ e method of successive bisection: we denote by / A the left or the right half of 7i according as 2 a n x n diverges or converges at the middle point of / . By the same rule, we designate a particular half of 7* by 7 a > anc * so on - The intervals of this nest (7 n ) all have the property that 2 a n x n converges at their left end point (say x n ) but diverges at their right end point (say yj. The number r (necessarily positive), which this nest determines, is the number required for the theorem. In fact, if x otf is any real number for which | a/ 1 < r (equality excluded), then we have | x' \ < x k , for a sufficiently large k, i. e. such that the length of J^ is less than r | a/ 1 . By theorem 1, xf is a point of convergence at the same time as x k is; and indeed at of we have absolute convergence. If, on the contrary, x" is a number for which | " | > r, then | x? \ > y tn , provided m is large enough for the length of 7 m to be less than | x" \ r . By theorem 2, x? is then a point of divergence at the same time as y m is. This proves all that was desired. This proof, which appeals to the mind by its extreme simplicity, is yet not entirely satisfying, in that it merely establishes the existence of the radius of convergence without supplying any information as to its magnitude. We will therefore prove the fundamental theorem by an alternative method, this time obtaining the magnitude of the radius itself. For this purpose, we proceed quite independently of our previous theorem, to prove the moie precise 14. Theorem 4 : // the power series 2 a n x n is given and JLI denotes the upper limit of the (positive) sequence of numbers i. 4 Cauchy: Analyse alge'brique p. 151. This beautiful theorem remained for the time entirely unnoticed, till J. Hadamard (J. de math, pures et appl., (4) Vol. 8, p. 107. 1892) rediscovered it and made use of it in important appli cations. g 18. The radius of convergence. 155 then a) if p, = 0, the power series is everywhere convergent; b) */ p. =r -{- oo, the power series is nowhere convergent; c) if < p < + oo , Me? power series converges absolutely for every |#|< > but diverges for every \ x \ > . Thus with the suitable interpretation, s tf& radius of convergence of the given power series*. Proof. If in case a) x is an arbitrary real number 4* 0, - r > and therefore by 59, 2 1 # 1 r for every n > m. By 87, 1, this shows that 2 a n x Q n converges ab- solutely, which proves a). If conversely 2 a n x n converges for x = x^ 4 s 0, then the sequence (a n x l n ) and, a fortiori, the sequence \V\a n x^\), are bounded. If n . n j y\ a n x" | < K 9 say, for every , then V| a n | <C i *-r = /C, for every n> L e. \V | a n | J is a bounded sequence. In case b), in which the sequence is assumed unbounded above, the series therefore cannot converge for any x =fc 0. Finally, in case c), if #' is any number for which | d \ < , then choose a positive Q for which |o;'| < Q < , and so > p,. By the definition of p, we must have, for every n > some n , Via I < and consequently I/I atf n \ < < 1. I nl g 1 i n i g By 75,1, 2 a n vd n is therefore (absolutely) convergent. ) For convenience of exposition, we here exceptionally write - = -f oo , : 0. Furthermore it should be noticed that is not for instance -f oo lim V| a n | _ i the same as lim , as the student should verify by means of obvious i/KT examples. (Cf. Ex. 24.) 156 Chapter V. Power series. On the other hand, if | x" \ > , so that -^ < ^, then we must have, for an infinite number of w's (again and again; v. 59) By 82, Theorem 1, therefore the series certainly cannot con- verge 6 . Thus the theorem is proved in all its parts. Remarks and Examples. 1. Since the three parts a), b), c) of the preceding theorem are mutually exclusive, it follows that the conditions are not merely sufficient, but also necessary for the corresponding behaviour of a n x n . n. 2. In particular, we have y | a n \ > for any power series everywhere convergent. For by the remark above, ^ = 0, and since we are concerned with a sequence of positive numbers, these certainly have their lower limit x^>fi. Since on the other hand must be < /* , we must have #=^ = 0. By 63 the sequence (|/ | a n \ j is therefore convergent with limit Thus for instance --0, or ywT->OO, n ' x* because 5? converges everywhere. (Cf. 43, Example 4.) Tjl 3. Theorems 93 and 94 give us no information as to the behaviour of the series for x= + r and for x = r; this differs from case to case: x tl , x n x n y. , y\ - all have the radius 1 . The first converges neither at 1 nor at n n* 1 , the second only at one of the two, the third at both. 4 Further examples of power series will occur continually in the course of the next paragraphs, so that we need not indicate any particular examples here. We saw that the convergence of a power series in the interior of the interval of convergence is, indeed, absolute convergence. We proceed to show further that the convergence is so pronounced as to be undisturbed by the introduction of decidedly large factors. We have in fact the CO 95. Theorem. // a n x n has the radius of convergence r, then the n=o GO ^ 00 power series ^na n x n " 1 t or what is the same thing, n=0 n=0 has precisely the same radius. 6 Case c) may be dealt with somewhat more concisely: If linTv^l n I = /x, then lim $\ ^T* n ~l - lim V^nl ' I * I =- M ' I * I (for what reason?). By 76, 3 the series is therefore absolutely convergent for fi | x | < 1, and certainly divergent for /x | x \ > 1, q. e. d. 18. The radius of convergence. 157 Proof. This theorem may be immediately inferred from Theo- rem 94. For if we write na n = a n ', then n - n, n. V I ' I = V M -V. Since (by 38, 5), ty~n+l> it follows at once from Theorem 62 that the sequences (y' I a ' |) and (ty I a M have the same upper limits. For if we pick out the same sub-sequences from both, as corresponding terms only differ by the factor y^, which * -)- 1, these sub-sequences either both diverge or both converge to the same limit 7 . Examples. 1. By repeated application of the theorem, we deduce that the series Sna n x n -*, 2n(n-V)a n x n ~*, ...,2n(n-l) (n k -f- or, what is exactly the same thing-, the series all have the same radius as 2a n x n , whatever positive integer be chosen for h. 2. The same of course is true of the series n * +a ? 2> Thus far we have only considered power series of the form a n x n . These considerations are scarcely altered, if we take the more general type n=0 Putting x x = x* , we see that these series converge absolutely foi but diverge for | x X Q \ > r , if r again denotes the number deter- mined by Theorem 94. The region of convergence of this series except in the extreme cases, in which it converges only for x = x , or for every x, is therefore a stretch bisected by the point # , sometimes with, sometimes without one or both of its end- points. Except for this displacement of the interval of convergence, all our considerations remain valid. The point # will for brevity be called the centre of the series. If X Q = , we have the previous form of the series again. 7 Alternative proof. By 76, 5a or 91, 2, the series 2nd*-* is convergent for every | & \ < 1. If | x \ <^ r, and Q is so chosen that I #o I "^ 6 "^ r * then 2a n Q n converges, (a n g n ) is therefore bounded, say We infer that K 6 e [ I , proves the convergence. which, since 158 Chapter V. Power series. In the interval of convergence, the power series S a n (x x ) n has a definite sum $, for each x, and usually of course a different sum for a different x. In order to express this dependence on x, we write and say that the power series defines, in its interval of convergence, a function of x. The foundations of the theory of real functions, that is to say the foundations of the differential and integral calculus, we assume, as remarked in the Introduction, to be already known to the reader in all that is essential. It is only to avoid any possible uncertainty as to the extent of the facts required from these domains, that we shall rapidly indicate, in the fol- lowing section, all the definitions and theorems which we shall need, without going into more exact elucidations or proofs. 19. Functions of a real variable. Definition 1 (Function). If to each value x of an interval of the #-axis, by any prescribed rule, a definite value y is made to correspond, then we say that y is a function ofx defined in that interval and write, for short, y =/(*), where "/" symbolises the prescribed rule in virtue of which each x has corresponding to it the relevant value of y. The interval, which may be closed or open on one or both sides, bounded or unbounded, is called the interval of definition of f(x). Definition 2 (Boundedness). If there exists a constant K such that for every x of the interval of definition we have then the function f(x) is said to be bounded on the left (or below) in the interval, and K 1 is a bound below (or left hand bound) of /(#). If there exists a constant K 2 such that for every x of the interval of definition / (x) ^ K 2 , then f(x) is said to be bounded on the right (or above) and K 2 is a bound above (or right hand bound) of f(x). A function bounded on both sides is said simply to be bounded. There then exists a constant K such that for every x of the interval of definition, we have 19. Functions of a real variable. 159 Definition 3 (Upper and lower bound, oscillation). There is always a least one among all the bounds above of a bounded function, and always a greatest among all its bounds below 8 . The former we call the upper bound, the latter the lower bound, and their difference the oscillation of the function f(x) in its interval of definition. Corresponding desig- nations are defined for a sub-interval a' ... V of the interval of definition. Definition 4 (Limit of a function). If is a point of the interval of definition of a function /(#), or one of the endpoints of that interval, then the notation = c or f(x) > c for x -> f means that a) for every sequence of numbers x n of the interval of definition which converges to , but with all its terms different from , the sequence of the corresponding values ^n ==/(*) (=1, 2,3,,..) of the function converges to c; or b) an arbitrary positive number e being chosen, another positive number 8 -= 8 (e) can always be assigned, such that for all values of x in the interval of definition with | x - | < 8 but * 4= , we have 9 \f( X )-C\< S . The two forms of definition a) and b) mean precisely the same thing. Definition 5 (Right hand and left hand limits). If, in the case of definition 4, it is stipulated besides that all points x n or x taken into account lie to the right of f (which must not of course be the right hand endpoint of the interval of definition of / (#)), then we speak of a right hand limit (or limit on the right) and write lim/(*) = c; x->t + Q similarly we write and speak of a left hand limit (or limit on the left), if f is not the left hand endpoint of the interval of definition of /(#), and if points x n or x to the left of are alone taken into account. 8 Cf. 8, 2, and also 62. 9 The older notation \imf(x) for lim/(#) should be absolutely discarded since the whole point is that x is to remain 4= f. 160 Chapter V. Power series. Definition 5 a (Further types of limits). Besides the three types of limit already defined, the following may also occur 10 : lim / = | or > c , -f- oo , oo /(*)-) with one of the five supplementary indications ("motions of x") for #->, *f+0, *f 0, +-\-oo, oo. With reference to 2 and 3 there will be no difficulty in formulating precisely the definitions in the form a) or b) which correspond to the definitions just discussed. Since, as remarked, ue assume these matters to be familiar to the reader, in all essentials, we suppress all elucidations of detail and examples, and only emphasize that the value c to which a function tends, for instance for x * , need bear no relation whatever to the value of the function at . Only for this we will give an example: let f(x) be defined for every x by putting f(x) = if x is an irrational number, but f(x) = if a; is a rational number which in its lowest terms is of the form - (q >> 0) . Thus e. g. f () 0,etc. Here we have for every f For if s is an arbitrary positive number and m is so large that <I e , then in there are not more than a finite number of rational points whose (least posi- tive) denominator is < m . These we imagine marked in the interval 1 ...+! As there are only a finite number of them, we can find one nearest of all to ; (if f itself is one of these points we of course should not take it into account here). Let d denote its (positive) distance from f. Then every x t for which 0<|as-| <d, is either irrational, or a rational number whose least positive denominator q is > m . In the one case, / f (a;) = 0; in the other, = <; <>. Therefore we have, for every xinQ<^\x \<Z.d, *-0<. i. e., as asserted, If therefore f is in particular a rational number, then this limit differs decidedly from the value f(g) itself. Calculations with limits are rendered possible by the following theorem: 10 In the first of these three cases we say that f(x) tends or converges to c\ in the second and third cases: f(x) tends or diverges (definitely) to-f-QO or 00; and in all three, we speak of a definite behaviour or also of a limit in the wider sense. If f(x) shows none of these three modes of behaviour, then we say that / (x) diverges indefinitely for the motion of x under consideration. 19. Functions of a real variable. 161 Theorem 1. If f (x) , f% (x) , . . . f (x) are given functions (p some determinate positive integer), each of which, for one and the same motion of x of the types mentioned in Definition 5 a, tends to a finite bmit, say (a?) > ^ , . . . , f p (*) -> c p , then a) the function / = fc() + /;(*) + 4- /;(*)] 'x4- , 4- 4- v b) the function / = [A (*)/;(*)/;(*)] -*vv--v c) in particular, therefore, the function a f^ (x) > a c l , (a = arbitrary real number) and the function f (x) f^ (x) * c x c 9 ; d) the function ^-r-c * , provided c =f= . Theorem 2. If hm f (x) = c (4= cx>) , then /* (x) is bounded in a *->*; neighbourhood of f, i. e. two positive numbers d and K exist such that and corresponding statements hold in the case of a (finite) limf(x) for z * f + , f , + oo , oo. Definition 6 (Continuity at a point). If f is a point of the interval of definition of f (x) , then f(x) is said to be continuous at f if l\mf(x) exists and coincides with the value /*() of the function at limf (x) = If we include the definition of lim in this new definition, we may also state: Definition 6 a. f(x) is said to be continuous at a point , if for every sequence of x n 's of the interval of definition, which tends to f , the corresponding values of the function Definition 6b. f(x) is said to be continuous at , if, having chosen an arbitrary e > 0, we can always assign 8 = 8 (e) > 0, such that for every x of the interval of definition with |*-f|<8 we have |/(*)-/tf)|< Definition 7 (Right hand and left hand continuity). / (x) is said to be continuous on the right (right-handedly) or on the left (left-hand- edly) if lim/(#) exists at least for#->f-f-0or x -> g respectively, and coincides with /(). Corresponding to Theorem 1 we have here the Theorem 3- If AC*), AC*), /j> (#) are given functions (/> a 162 Chapter V. Power series. particular positive integer), all continuous at , then the functions )/!(*)+/ (*)++/,(*), b) /i (*) /, (*).../,(*), c) a/! (x) (a = an arbitrary real number), / x (#) / 2 (#), and d) if A () 4=0, also ^ are all continuous at . Corresponding statements hold, when only right hand or only left hand continuity is assumed. By repeated application of this theorem to the function f(x) ~ #, certainly continuous everywhere (since for x -> we have precisely x -> ), we at once deduce: All rational functions are continuous everywhere, with the exception of (at most a finite number of) points where the denominator = 0. In particular: Rational integral functions are continuous everywhere. Similarly, the limiting relations 42, 1 3, showed that: a x y (a > 0) is continuous for every real x\ log x is continuous for every x > 0; x* (a ~ arbitrary real number) is continuous for every x > 0. Definition 8 (Continuity in an interval). If a function is con- tinuous at every individual point of an interval /, then we say that it is continuous in this interval. Continuity at an endpoint of the interval is here taken to be continuity "inwards", i. e. right handed continuity at the left hand endpoint, and left handed continuity at the right hand end- point. These endpoints of/ may or may not, according to the circum- stances, be reckoned as in the interval. Functions which are continuous in a closed interval give rise to a series of important theorems, of which we may mention the following: Theorem 4. If f(x) is continuous in the closed interval a ^ x ^ b and if f(a) > 0, but f(b) < 0, then there exists, between a and b, at least one point f for which /() = 0. Theorem 4a. If f(x) is continuous in the closed interval a ^ x ^ b and 77 is any real number between /(a) and/ (b), then there exists, between a and b, at least one point f for which/ () = 17. Or: The equation/ (x) -= 17 has at least one solution in that interval. Theorem 5. If f(x) is continuous in the closed interval a ^ x ^ b y then, having chosen any e > 0, we can always assign some number 8 > so that, if x' and x" are any two points of the interval in question whose distance | x" x' \ is < 8, the difference of the corresponding values of the function, | /(*") /(#') | , is < e. (The property, established by this theorem, of a function continuous in a closed interval is called uniform continuity of the function in the interval.) Definition 9 (Monotony). A function defined in the interval a ... b is said to be monotone increasing or decreasing in the interval, if for every pair of points x l and # 2 f that interval, with x l < # 2 , we in- 19. Functions of a real variable. 163 variably have f (x ) < f(x^) in the one case, or invariably f(x t ] in the other. We also speak of strictly increasing and strictly de- creasing functions, when the equality signs, in the inequalities between the values of the function just written down, are excluded. Theorem 6. The point f , certainly existing under the hypotheses of Theorems 4 and 4 a, is necessarily unique of its kind if the func- tion f(x) under consideration is strictly monotone in the interval a... b. Thus in that case, to each 77 between f(d) and f(b) corresponds one and only one for which /*() = rj. We say in this case: The inverse function of y = f(x) is everywhere existent and one-valued (or y = f(x) is reversible) in the interval. Definition 10 (Differentiability). A function f (x) defined at a point | and in a certain neighbourhood of f is said to be differentiable at * if the limit exists. Its value is called the (unique derivative or) differential coefficient of f(x) at and is denoted by /"'(). If the limit in question only exists on the left or on the right (that is, only for x *f-f-0 or x f respectively), then we speak of right hand or left hand differentiability, differential coefficient, etc. If a function is differentiable at each individual point of an inter- val /, then we say for brevity that the function is differentiable in this interval. The rules for differentiation of a sum or product of a particular (fixed) number of functions, of a difference or quotient of two functions, of functions of a function, as also the rules for differentiation of the elementary functions and of their combinations, we regard as known to the reader. All means necessary to their construction have been developed in the above, if we anticipate a knowledge of the limit defined in 1155 and there determined in a perfectly direct manner. If, for instance, it is inquired whether a x (a> and ={= 1) is differentiable, and, if so, what is its differential coefficient, at the point f , then, following Defs. 10 and 4, we have to choose a null se- quence (#) with terms all =f= and to examine the sequence of numbers __ **+*- a* a r -l A. = -- :_= a s - . Xn X n If we write y n for the numerator in the last fraction, then by 35, 3 we know that (y n ) is also a null sequence, and indeed one for which none of the terms is equal to 0. X n may then be written in the form J But since, as remarked, y n is a null sequence, we have by Since the same then holds for the reciprocal values, by 41, 11 a, we deduce A^-x^.log-fl. The function a* is thus differentiable for every x and has the differential coefficient a r -loga. 164 Chapter V. Power series. In the same way, as regards differentiability and differential coefficient of log x for f !> 0, we deduce, by consideration of x n x n that the differential coefficient exists here and = -- . Of the properties of differentiable functions we shall for the pre- sent require scarcely more than is contained in the following simple theorems : Theorem 7. If a function f(x) is differentiable in an interval J and its differential coefficient is there constantly equal to 0, then f(x) is constant in /, that is to say is = /*(a? ), where X Q is any point of /. If two functions f t (x) and f^ (x) are differentiable in /and their differential coefficients constantly coincide there, then the difference of the two functions is constant in /, therefore we have &(*)- /i(*)+' = /i(*) + [/;(*o)-/;(*o)] where x is any point of /. Theorem 8. (First mean value theorem of the differential calculus.) If f(x) is continuous in the closed interval a ^ x <^ b and differentiable in at least the open interval a < x < b, then there is, in the latter interval, at least one point f for which (In words: The finite difference quotient relative to the endpoints of the intervals is equal to the differential coefficient at a suitable interior point.) Theorem 9. If /(*) is differentiable at and/' () is > (< 0) then /(#) "increases" ("decreases") at , i. e. the difference .// \ ^//-\i_ ( * ne same ) . /*. \ / r\ /(*)-/() has . s,gn as (to) (* - & provided | x | be less than a suitable number 8. Theorem 10. If f(x) is differentiable at an interior point of its interval of definition, then unless /' () = the functional value /() cannot be ^ every other functional value f(x) in a neighbourhood of of the form | x | < 8, i. e. cannot be a (relative) maximum point. Similarly the condition /' (f ) = is necessary for to be a (relative) minimum point, i. e. such that/() is not greater than any other functional value /(#), as long as x remains in a suitable neighbourhood of . Definition 11 (Differential coefficients of higher orders. If f(x) is differentiable in J y then (in accordance with Def. 1) /' (x) is again a function defined in / *. If this function is again differentiable in J l 1 and called the derived function of f(x). 19. Functions of a real variable. 165 then its differential coefficient is called the second differential coefficient of f(x) and is denoted by /"'(#). Correspondingly, we obtain the third and, generally, the ft th differential coefficient of f(x), which is denoted by f (k) (x). For the existence of the k th differential coefficient at f it is thus (v. Def. 10) necessary that the (k l) th differential coefficient should exist both atf and at all points of a certain neighbourhood of . The J th differential coefficient of f^(x) is f( k +*(x), k^O, l^>0. (As th differential coefficient of f(x) we then take the function itself) Of the integral calculus we shall, in the sequel, require only the simplest concepts and theorems, except in the two paragraphs on Fourier series, where rather deeper material has to be brought in. Definition 12 (Indefinite integral). If a function f(x) is given in an interval a ... b and if a differ entiablc function F (x) can be found such that, for all points of the interval in question, F f (x) = f(x), then we say that F(x) is an indefinite integral of f(x) in that interval. ( Be- sides F(x), the functions F(x) -\- c are then also indefinite integrals of f(x), if r denotes any real number. Besides these, however, there are no others). We write In the simplest cases, indefinite integrals are obtained by inverting- the elementary formulae of the differential calculus E. g. from (sin a x)' = a cos ax it follows that \cosccxdx = -- and so on These elementary rules we J a assume known Special integrals of this kind, excepting- the very simplest, are little used in the sequel; we mention / Ji J[- /7 ' 1 1 1 9 f 1 - H-x" 3 "* ^ ' ~' 6 ~" ^ ' " V 3" V3 = V^ log *1^][A + V^ [tan- 1 (A: V 2 - 1) + tan" 1 (* V 2 -f 1)], 4 o 3.2 _ ~ . / f> _, i 4 j t cot x dx = log Though in indefinite integrals, we find no more than a new mode of writing for formulae of the differential calculus, the definite integral introduces an essentially new concept. Definition 13 (Definite integral). A function defined in a closed interval a ... b and there bounded is said to be integrable over this interval if it fulfils the following condition: Divide the interval a ... b in any manner into n equal or un- equal parts (n ^> 1, a positive integer), and denote by x l9 a? a , . . . , # n _ the points of division between a = X Q and b = x n . Next in each of these n parts (in which both endpoints may be reckoned) choose any 166 Chapter V. Power series. point, and denote the chosen points in corresponding order by g l9 f 2 , . . . , g n . Then form the sum 11 S n = 2? (*-*_.!)/(&) i/-=i Let such sums S n be evaluated for each n = 1, 2, 3, ... independently (that is to say, at each stage x v and , may be chosen afresh). But, at the same time, / n , the length of the longest of the n parts into which the interval is divided when forming S ny shall tend to 12 . If the sequence of numbers S l9 S 2 > . . . , in whatever way they may have been formed, invariably proves to be convergent and always gives the same 13 limit S, then f(x) will be called integrable in Riemann's sense and the limit S will be called the definite integral of f(x) over a . . . b, and written }f(x)dx. a x is called the variable of integration and may of course be replaced by any other letter. Instead of /() we may also take, to form S n , the lower bound <z v or the upper bound f3 v of all the functional values 14 in the interval #,.__! ^ x ^ x v . Theorem 1 1 (Riemann's test of integr ability). The necessary and suffi- cient condition for a function /(#), defined in the closed interval a ... b and there bounded, to be integrable over a . . . b, is as follows: Given e > 0, a choice of n and of the points x l9 x 2 , . . . , x n _ l must be possible, for which if i v = | x v A:,,_ I | 5s the length of the v th part of a ... b and v v the oscillation of f(x) in this sub-interval. This criterion may also be expressed as follows, assuming the notation chosen so that a < b: After choosing e, we must be able to assign two "step-functions" (functions constant in stretches) such that in a ^ x ^ b we have always *(*)^ /(*)<<?(*) 11 If /(*) > 0, a > b, and we consider a plane portion S bounded on the one side by the axis of abscissae, on the other by the verticals through a and b and by the curve y f (x) t then S n is an approximate value of the area of S. This however only provides a satisfactory representation if y = / (x) is a curve in the intuitive sense. 12 We may then also say that the subdivisions, with increasing n t become indefinitely closer. 13 It is easily shewn that if the sequence (S n ) is invariably convergent it also ipso facto always gives the same limit. 14 In these cases S n gives the area of a ("step-*') polygon inscribed or circum- scribed to the plane portion S. 19. Functions of a real variable. 167 b as well as 15 J (G (x) g (x)) d x < e. a It suffices in fact to put, in x v _ : ^ x ^ #, g (x) -= a,, G (x) = p v , v = 1, 2, . . . , w, together with (b) = a n , G (ft) - n . From this criterion, the following particular theorems are deduced: Theorem 12. Every function monotone in a ^ # ^ ft, and also every function continuous in a f^ # 5^ ft, is integrable over a ... ft. Theorem 13. The function/ (#) is integrable over a ... ft, if, in a ... ft, it is bounded and has only a finite number of discontinuities. Riemann's test of integrabiiity may also be given the following form: Theorem 14. The function f(x) is integrable over a ... ft if, and only if, it is bounded there and if, two arbitrary positive numbers 8 and e being assigned, the subdivision of a ... ft into n sub-intervals described in theorem 11 can be so carried out that the sub-intervals i v in which the oscillation of f(x) exceeds 8 add up to a total length less than e. Theorem 1 5. The function /(#) is certainly not integrable over a ... ft if it is discontinuous at every point of that interval. Theorem 16. If f(x) is integrable over ... ft, then/(,v) is also in- tegrable over every sub-interval a' ... ft' of a ... ft. Theorem 17. If the function /(jc) is integrable over a ... ft, then every other function f (x) is integrable over ... ft, and has the same integral, which results from/(.r) by an arbitrary change in a finite number of its values. Theorem 18. If f(x) and / x (x) are two functions integrable over a ... ft, then they have the same integral provided that they coincide at least at all points of a set everywhere dense in ... ft (e. g. all rational points). For calculations with integrals we have the following simple theorems, where /(*) denotes a function integrable over the interval ... ft. a b Theorem 19. We have lf(x)dx --- lf(x)dx and if a l9 2 > a s b a are three arbitrary points of the interval ... ft, ?/(*) dx + ff(x)dx + ]f(x)dx - 0. <*i a\ <** Theorem 20. Iff(x) and# (x) are two functions integrable over ... ft, ( < 6), and if in ... ft we have constantly f(x)^g (x), then we also have b b It is immediately obvious from the first form of the criterion that a step- function such as G (x) g (x) is integrable. 168 Chapter V. Power series. Theorem 20a. \f(x) | is integrable with/(#) and we have, if a < b, | }/(*)</*! J|/(*) | </*. a a Theorem 21. (First mean value theorem of the integral calculus?) We have b if IJL is a suitable number between the lower bound a and the upper bound j8 of f(x) in a . . . b (a 5g ju, fg j8). In particular we have | //(*)<**! ^ *(*-) a if K denotes a bound above of \f(x) | in a ... b. Theorem 22. If the functions / x (#),/ 2 (#), . . . t f p (x) are all integrable over a . . . b (p = fixed positive integer), then so are their sum and their product and for the integral of the sum we have the formula J (A (*) + +/P (*))** = J/i (*)/*+...+ J/p (*)/*; a i. e. the sum of a jfixa/ number of functions may be integrated term by term. Theorem 22a. If f(x) is integrable over a . . . b and if the lower bound of | f(x) \ in a . . . b is > 0, then ^ is also integrable over a ... b. J ( x ) Theorem 23. If f(x) is integrable over a . . . b, then the function is continuous in the interval a . . . b and is also differentidble at every point of the interval, where /(^c) itself is continuous. If X Q is such a point, then *"(*Q)=/(*O) there - Theorem 24 (Fundamental theorem of the differential and integral calculus). If/(*) is integrable over a ... b, and has an indefinite integral -F (#) in that interval, then Theorem 25 (Change of the variable of integration) . If f(x) is integrable over a ... b and x = 9 (J) is a function diffcrentiable in a . . . /3, with 9 (a) = and 9 (ft) = b, if further, when t varies from a to /?, 9 (/) varies monotonely (in the stricter sense) from a to 4, and if 9' (/), the differential coefficient of 9 (J), is integrable 16 over a . . . /?, then //(*)<**=//(?(<)) 9' <*' a a 10 The derivative of a differentiable function need not be integrable. Examples of this fact are, however, not very easily constructed (cf. e. g. H. Lebesgue, Leyons sur 1' integration, 2 nd Edition, Paris 1928, pp. 9394). 19. Functions of a real variable. 169 Theorem 26 (Integration by parts). If/(#) is intcgrable over a . . . b and F (x) is the indefinite integral of /(#), if further g (x) is a function, differentiate in a . . . b, whose differential coefficient is inte- grablc over a . . . b, then 17 //(*) g(*)dx=[F (*) i- (*)].* - } F (x) g' () rf *. a a The following penetrates considerably further than all the above simple theorems: Theorem 27 (Second mean value theorem of the inte- gral calculus). If / (x) and 9 (x) are integrable over a ... b and 9 (x) is monotone in that interval, then a number , with a ^ ^ 6, can be so chosen that J 9 (*) /(*)</* -= 9 () \f(*)dx + 9 (ft) }/(*) d x. a a $ Here 9 (a) may also be replaced by the limit, certainly existing under the hypotheses, 9 a = Iim9(#), and similarly 9 (b) by o b lim 9 (#); but in *-a+0 ->6-fO this case a different value may have to be chosen for . We mention only the following of the applications of the concept of integral above considered: Theorem 28 (Area). Iff(x) is integrable over a . . . b, (a < b) and, let us suppose, always positive in the interval 18 , then the portion of plane surface bounded by the axis of abscissae, the ordinates through a and b, and the curve y / (x) or more precisely, the set of points (.v, y) for which a ^ x ^ b, and at the same time, for each such x, r y ^f(x) 9 b has a measurable area and its measure is //(#) d x. a Theorem 29 (Length) . If x = 9 (t) and y = ip (t) are two functions differentiable in a ^ ^ /?, and if 9' () and </'' (0 themselves are con- tinuous in a ... j8, then the path traced out by the point x = 9 (t), y = /> (/) in the plane of a rectangular coordinate system O x, O jy, when / describes the interval from a to /3, has a measurable length and this is given by the integral ~~' Finally we may say a few words on the subject of so-called improper integrals. 17 Here [7i (x)]* denotes the difference h (b) h (a). 18 which may always be arranged by the addition of a suitable constant. 170 Chapter V. Power series. Definition 14. If /(/) is defined for t^. a and is integrable over ^ t ~i x > f r every x, so that the function is also defined for every x ^ a, then, if lim jP (x) exists and = c, we say that the improper integral converges and has the value c. Theorem 30. If f(t) is constantly ^ or constantly .< for every 00 t ^ a, then //(*) d t converges if and only if the function F (x) of Def. 14 a is bounded for x > a. If f(i) is capable of both signs for t ^ a, then the same integral converges if, and only if, given an arbitrary e > 0, x > a can be so determined that for every x' and x" both > x . And quite analogously: Definition 15. If f(x) is defined, but not bounded, in the interval a < t <; b, open on the left, and is integrable, for every x of a <x <b, over the interval x ^ t ^ b, so that the function F(x)=*}f(t)dt X is defined for each of these x's, then, if lim F (x) exists and = c 9 we say A.-^ + that the improper integral (improper at a) is convergent and has the value c. Exactly analogous conventions are made for an interval open on the right. The case of an interval open on both sides is reduced to the two pre- ceding cases by dividing it at an interior point into two half-open intervals, and then taking theorem 19 as a definition. Theorem 31 . If in the case of Def. 15, we further have/(f) S> every- where or ^ everywhere, then the improper integral in question exists if, and only if, F (x) remains bounded in a < x ^ b. If f(t) assumes both signs, then the integral exists if, and only if, given e > 0, we can choose 8 > so that for every x' and x" both between a (excl.) and a + 8. 20. Principal properties of functions represented by power series. 171 20. Principal properties of functions represented by power series. We interrupted our discussion of power series at the observation, terminating 18, that the sum of a power series, in the interior of its interval of convergence, defines a function, which we will now denote by f(x): We resume it at that point, and agree in this connection, unless special remark to the contrary is made, to leave the interval of con- vergence open at both ends, even should the power series converge at one or both of the endpoints. Now if, as is the case here, an infinite series defines a function in a certain interval, then the most important problem is, in general, to deduce from the series the principal properties of the function re- presented by it interpreting these for instance in the sense of the summary of the preceding section. In the case of power series, this presents no great difficulty. We shall see, on the whole, that a function represented by a power series possesses all the properties which we may consider particularly im- portant and that the algebra of power series assumes a peculiarly simple form. For this reason, power series play a prominent part, and it is precisely on this account that their discussion belongs to the elements of the theory of infinite series. In these investigations, we may, without thereby restricting the scope of the results, assume X Q = 0, i. e. assume the series to be of the simplified form 2 a n x n . Its radius of convergence is of course assumed positive (> 0), but may be + oo, i. e. the series may be everywhere convergent. We then have, first, the Theorem. The function f(x) defined, in its interval of conver- 96 gence, by the power series a n (x x ( j) n , is continuous at x = x ; n=o that is to say, we have lim f(x) = lim J a n (x - xj = * = f(x Q ). n=0 Proof. If < Q < r , then by 83, 5, 00 00 J^Kik n ~ l converges with J|0j n . n=l n=o If we write K(> 0) for the sum of the former, then we have, for every | x x \ < g, \f(*)- Q \ = \(*-*o)'2n(x-*o) n - l \^\* -*<>]*< n=*l 172 Chapter V. Power series. If therefore e > is arbitrarily given and if d > is less than both Q and ;, then we have, for every | x X Q \ < 5, l/-ol<; which by 19, Def 6b, proves all that was required. From this theorem, we immediately deduce the extremely far- reaching and very frequently applied: 97. Identity Theorem for power series. // the two power series 2a n x n and ^b n x n n-o n-O have the same sum in an interval \ x\ < Q in which both of them converge l9 , then the two series are entirely identical, that is to say, for every n = 0, 1, 2, . . , we then have n = b n . Proof From (a) a + ^ x + a 2 * 2 -\ = b + 6 x + 6 3 x* ^ it follows, by the preceding theorem, letting x+0 on both sides of the equation, that o ^ V Leaving out these terms and dividing by x t we infer that for < \x\ < Q (b) *!+*** + ****{ =& 1 +& 2 * + &a a;9 H , an equation from which we deduce, in exactly the same way 20 , that *i = & i and ao+^aH = & 9 + 63^ Proceeding in this manner, we infer successively (more precisely: by complete induction) that for every n the statement is fulfilled. Examples and illustrations. 1. This identity theorem will often appear both in the theory and in the applications. We may also interpret it thus: if a function can be re- presented by a power series in the neighbourhood of the origin, then this is only possible in one way. In this form, the theorem may also be called the theorem of uniqueness It of course holds, in the corresponding statement, for the general power series 2a n (x x ) n . 2. Since the assertion in the theorem culminates in the fact that the corresponding coefficients on both sides of the equation (a) are equal, we may also speak, when applying the theorem, of the method of equating coefficients. 19 Or even for every x = x v of a null sequence (x v ) whose terms are all =fc 0. In the proof we have then to carry out the limiting processes in accordance with 19, Def. 4a. 20 For x = 0, equation (b) is not in the first instance secured, since it was established by means of division by x. But for the limiting process x -> this is quite immaterial (cf. 19, Def. 4). 20. Principal properties of functions represented by power series. 173 3. A simple example of this form of application is the following: We certainly have, for every a;, v=0 If we multiply out on the left, by 91, Rem. 1, and equate the coefficients on both sides, then we obtain, for instance, by equating- the coefficients of x k : a relation between the binomial coefficients which would not have been so easy to prove by other methods. 4. If f (x) is defined for | x \ < r and we have, for all such a;'s, /(-*>-/(*). then f(x) is called an even function. If it is representable by a power series, then we at once obtain by equating 1 coefficients, so that in the power scries of f(x), only even powers of x can have coefficients different from 0. 5. If on the other hand, f(x) = f(x), then the function is said to be odd. Its expansion in power series can then only contain odd powers of x. In particular, /"(O) = 0. We now proceed one step further and prove a number of theorems which must be regarded as in every respect the most important in the theory of infinite series: Theorem 1. // 08. n=0 is a power series with (positive) radius r, then the function f(x) thereby represented, for \x X Q \ < r f may also be expanded in a power series with any other point x of the interval of convergence as centre; we have, in fact, *> where *= and the radius r of this new series is at least equal to the positive number r | x X Q \ . Proof. If x l lies in the interval of convergence of the series, so that | #! x 1 < r, then i. e. = 174 Chapter V. Power series. and all that we have to show is that we may here group together all terms with the same power of (x &J, i. e. that the main re- arrangement theorem 9O may be applied. If, however, to test its validity, we replace, in the latter series, every term by its absolute value, then we obtain 'the series n=0 and this is certainly still convergent, if K - *o I + I * - *i I < * > or I - *i I < ' - K - *o I If therefore x is nearer to # t than either of the endpoints of the original interval of convergence, then the projected rearrangement is allowed, and we obtain for f(x), as asserted, a representation of the form If we proceed in detail to group the terms containing (x - #,) fe to- gether, by writing the terms of the series (a) in successive rows one below the other, then the k th column gives which completes the required proof 21 . From this theorem we deduce the most diverse consequences. First we have the Theorem 2. A function represented by a power series /(*) = Jx(*-*o) n n=o is continuous at every point x^ interior to the interval of convergence. Proof. By the preceding theorem, we may write, for a certain neighbourhood of x , /(*)- Jk(*-*o)" = 2*.(*-xf n=0 n=0 with & = J}* n (*i-*oT = f(*i)- rt=0 For x *a? , the second of the representations of f(x), by 96, at once gives the required relation (v. 19, Def. 6): lim f(x) = f(xj. Theorem 3. A function represented by a power series n=0 is differentiable at every interior point x of the ' interval of convergence 21 We thus have, quite incidentally, a fresh proof of the convergence, already established in 95, of the different series obtained for the coefficients b k . 20. Principal properties of functions represented by power series. 175 (v. 19, Def. 10) and its differential coefficient at that point, f' '(xj, may be obtained by means of term-by-term differentiation, i. e. we have f fo) = n a n (x, - xjp-i = (n + 1) * w+1 (x, - x )\ n=l n=0 Proof. Since f(x) = 2Jb n (x xj", we have for every x suf- n=o ficiently near x^: whence for x >a; 1 , by 96, taking into account the meaning of b 19 we at once deduce the required result: f ( 1 ) = b l = 2 na n ( x i x o} n ~ l * Theorem 4. A function represented by a power series, f (*) = %*(* -*tf> n=o has, at every interior point x^ of its interval of convergence, differential coefficients of every order and we have /<*>(*,) = ^ = Z(n + !)( + 2) ... (n + k)a n + k (x, - * )"- n=o Proof. For every x of the interval of convergence we have, as we have just shown, f'(x) is thus again a function represented by a power series, and in fact by one which, in accordance with 95, has the same interval of convergence as the original series. Hence the same result may be again applied to f'(x}> giving r (x) = jj n (n + 1) * n+1 (x - * )i-i = 2(n + 1) (n + 2) * n + 2 (x - x ) n . n=i n=0 By a repetition of this simple process, we obtain for every k, valid for every x of the original interval of convergence. Putting in particular x = x l9 we therefore at once deduce the required statement. If we substitute, for the coefficients b k in the expansion of theorem 1, the values -j-\f (k) ( x i) now obtained, then we finally infer from all the above the so-called Taylor series 22 . // for \x x Q \<r, we have 99. /(*) = J: (*-*<,)"> n=o and if x is an interior point of the interval of convergence, then we 33 Brook Taylor: Methodus incrementorum directa et inversa, London 1715. Cf. A. Pnngsheim, Gcscliichte dcs Taylorschen Lehrsatzes, Bibl. math. (3) Vol. 1, p. 433. 1900. 176 Chapter V. Power series. have, for every x for which 23 | x x l \ < r l - r | x l X Q \, With Theorem 3 for the differentiation of our series, we couple the cor- responding theorem for integration. Since a function represented by a power series is continuous in the interior of its interval of convergence, it is also, by 19, Theorem 12, integrable over every interval contained, together with its endpoints, in the interior of this interval of convergence. For this we have the Theorem 5. The integral of the (continuous) function f (x) represented 00 by a n (x x ) n in the interval of convergence, may be obtained by means n-^Q of term by term integration, with the formula provided x t and x 2 are both interior to the interval of convergence. Proof. By 95, 2, the power series has the same interval of convergence as the given series =o By 98, 3, the first series is an indefinite integral of the second. Hence by 19, Theorem 24, the statement follows at once. These theorems on power series we may complete in a special direction by the following important addition: Theorem 2 on the continuity of the function represented by a power series was, as we may again expressly observe, only valid for the open interval of con- vergence. Thus, for instance, in the case of the geometric series 2! x n , 23 The number r l = r \ x l x \ of the text need not be the exact radius of convergence of the new series. On the contrary, the latter may prove considerably larger. Thus for f(x) S x n = ^ __ and x^ = - we obtain, by an easy cal- culation, .... /2\*+l / 1\* /W ^?o0 (-+) 1 and the radius of this series is not r | x^ x \ -= ^ but is ~. 20. Principal properties of functions represented by power series. 177 of sum ^ , we can deduce from our considerations neither its con- X ~ X tinuity at the point a; = 1, nor its discontinuity at rr=+l, by immediate inspection of the series. Even if the power series con- verged at one of the endpoints of the intervals (as here J for B = ij , we should not be able to conclude this fact directly. That however, in this last particular case, the presumption is, at least to some extent, justified, we learn from the following: Abel's limit theorem 24 . Let the power series f (x) = JJ a n x n 100. n=0 have radius of convergence r and still converge for x = -f- r . Then \imf(x) exists and = Ja n r n . a?->r n o 00 Or in other words: If 2,a n x n still converges for #= -fy, then n = the function f (x) defined by the series in r < x <^ -f- r , is also continuous on the left at the endpoint x = -f- r. Proof. There is no restriction 25 in assuming r = + 1. For if 2a n x n has radius r, then the series Za^'x", in which a n ' = a n r n , obviously has radius 1; and the latter series is convergent at + 1 or 1 , if, and only if, the former was at -j- r or r respectively. We therefore in future assume y = -J- 1 . Our hypothesis is, therefore, that f(x) = 2a n x n has radius 1 and that 2a n = s con- verges; and our statement is that 3D \[mf(x) = s, i.e. = a n . *->l-0 n=0 Now by 01 (v. also later, 102), we have for \x\ < 1, 1 00 00 00 00 * V a x n V rr n V a. rr n V * <r n 1 _ x ^ a n X ~ X 2j a n X ~ 2. S n X > 1 * n=o n =o n^O n-0 if by s n we denote the partial sum of 2 a n . Consequently f(x) = (l x] 2s n x n and since 1 = (1 x)2x n , we therefore deduce, for \ X \<1> (a) s - f(x] (1 - x)2(s - sj x n ~(l - x}^r n x\ n=0 =0 24 Journal f. d. reine u. angew. Math. Vol 1, p. 311. 1826. cf. 233 and 62. The theorem had already been stated and used by Gauss (Disquis. generalcs circa seriem ..., 1812; Wcrke III, p 143) and in fact precisely in the form proved further on, that r n -* involves (1 x) 2 r n x n -> if x * 1 from the left (v. eq. (a)). The proof given by Gauss loc. cit. is however in- correct, as he interchanged the two limiting processes which come under con- sideration for this theorem, without at all testing whether he was justified in so doing 26 This remark holds in general for all discussions of (not everywhere convergent) power series of positive radius f. 178 Chapter V. Power series. Here we have written s s n = r w , the "remainder" of the series; these remainders, by 82, Theorem 2, form a null sequence. If now e > is arbitrarily given, then we first choose m so large that, *" for every n > m y we have | r n \ < . We then have, for :g x < 1, 2t \s-f(x) I ^ I (1 - x) Zr n x\ + | (1 - x) -Z x, w^O * n tn + 1 hence, if /> denotes a positive number greater than | ^0 I + I r i I + + |r m |, this is e v' -r- 1 ^/ (!-*)+ J(l -*)?. If we now write 8 = the smaller of the two numbers 1 and - then we have, for 1 - 8 < x < 1, P> !-/(*)! <y+ J = , which, by 19, Def. 5, proves the required statement "/ (x) -> s for x -> 1 -0". We have of course, quite similarly, Abel's limit theorem for the left endpoint of the interval of convergence: X If E a n x n still converge for x = r, then w-=0 lim/(*) exists and = ^ ( l) n n r n . x-> r + w-0 The continuity theorem 98, 2 and Abel's theorem 100 together assert that 101. lim (Za n x n ) =2a n % n *-> if the series on the right converges and x tends to from the side on which lies the origin. If the series Ea n n diverges, we cannot assert anything, without further assumptions, as to the behaviour of Ha n x n when x -> g. We have however in this connection the following somewhat more definite: Theorem If 2a n is a divergent series of positive terms, and Z a n x n has radius 1, then /(*) = a n x^+ oo H = when x tends towards + 1 from the origin. Proof. A divergent series of positive terms can only diverge to + oo . If therefore G > is arbitrarily given, we can choose m so large that a + a^ + . . . + a m > G + 1, and then by 19, Theorem 3, choose 8 < 1 so small that for every 1 > x > 1 8, we continue to have a + a 1 x+ ... + a m x m > G. 21. The algebra of power series. 179 But then we have, a fortiori, /(*)= 2 a nX >G, n which is all that required proof. Remarks and examples for the theorems of the present paragraph will be given in detail in the next chapter. 21. The algebra of power series. Before we make use of the far-reaching theorems of the preceding section ( 20), which lead to the very centre of the wide field of application of the theory of infinite series, we will enter into a few questions whose solution should facilitate our operations on power series. That power series, as long as they converge, may be added and sub- tracted term by term already follows from 83, 3 and 4. That we may immediately multiply out term by term, in the product of two power series, provided we remain in the interior of the intervals of convergence, follows at once from 91, since power series always converge absolutely i/i the interior of their intervals of convergence. We therefore have, with also E * E b nX = E n n - 71 provided x is interior to the intervals of convergence of both series 26 . The formulae 91, Rem. 2 and 3 were themselves a first application of this theorem. If the second series is, in particular, the geometric series, then we find oo QO on 2 a n oc n E x n = E s n * n , n -0 H = O N---0 1 TO 00 i.e. - 2 a n x n = E s n x n A x w-0 n-O oo oo or E a n x n = (1 - x) E s n x n , 103 n n --- where s n = a Q -f- a 1 + + a n , and \ x \ < 1 and also less than the radius of E a n x n . We infer in as simple a manner that every series may be multiplied and in fact, arbitrarily often by itself. Thus ( E a n x n \ = E K a n + a l a n ^ + . . . + a n a Q ) X n ; Vi / n ^0 and generally, for every positive integral exponent k y !a n x n } k = E a n w** 108 - i-O / w-O 20 Here we see the particular importance of Cauchy's product (v. 91, 1). 180 Chapter V. Power series where the coefficients a ( are constructed from the coefficients a n in a perfectly determinate manner even though not an extremely obvious one 27 for larger yfe's. And these series are all absolutely convergent, so long as 2a n x n itself is. This result makes it seem probable that we "may" also divide by power series, that for instance we may also write and that the coefficients c n may again be constructed in a perfectly determinate manner from the coefficients a n . For we may first, writing -- " = ', for n = 1 , 2 , 3 , . . . , replace the left hand ratio by 1 _ and then by which must actually result in a power series of the form .Sc n x n , if the powers are expanded by 103 and like powers of x then grouped together. Our justification for writing the above may at once be tested from a somewhat more general point of view: We suppose given a power series 2a n x n (in the above, the series a n f a: n ) , whose sum we denote by f(x) or more shortly by y. n=0 We further suppose given a power series in y, for instance g (y) = 2 b n y n (in the above, the geometric series 2 y n ) and in this we substitute for y the former power series: b o + *i K + i x + ) + 6 9 K + a i x + 0* H Under what conditions do we, by expanding all the powers, in accordance with 103, and grouping like powers of x together, ob- tain a new power series C Q + c x -}- c a x 2 -) ---- which converges and has for sum the value of the function of a function g(f(x))? We assert the 104. Theorem. This certainly holds for every x for which converges and has a sum less than the radius of 2b n y n . n=0 27 Recurrence formulae for the evoluation of a are to be found in /. W. L Glaisher, Note on Sylvester's paper: Development of an idea of Eisen stein (Quarterly Journal, Vol. 14, p. 79 84. 1875), where further references to the bibliography may also be obtained. See also B. Hansted, Tidbkrift for Mathematik, (4) Vol. 5, pp. 1216, 1881. 21. The algebra of power series. 181 Proof. We have obviously here a case of the main rearrangement theorem 90, and we have only to verify that the hypotheses of that theorem are fulfilled. If we first write forming the powers by 103, and also suppose this notation 28 adopted for k = and k = 1, then we have, in (A) 6, y = b l (a (i) rfv- .<*'*. <*) x". the series z'* 1 ' occurring in the theorem 90. If we now take, instead of y = 2 a n x" , the series 97 = 2 \ a n x" \ , and, writing | x | = | , form, quite similarly, (A') then all the numbers in this array (A'j are ^ and since furthermore ^ I ^fe I *? fc was assume d to converge, the main rearrangement theorem is applicable to (A'), But obviously every number of the array A is in absolute value <^ the corresponding number in (A'); hence our theorem is a fortiori applicable to (A) (cf. 90, Rem. 3). In particular, therefore, the coefficients standing vertically one below the other in (A) always form (absolutely) convergent series 00 yjb k a^ k) = c n (for every definite n = 0, 1, 2, ...) and the power scries formed with these numbers as coefficients, i. e. is again, for the considered values of x, (absolutely) convergent and has the same sum as 2b n y n . We therefore have, as asserted, with the indicated meaning of n=0 Remarks and Examples. 105. 1. If the "outer" series gy) = ^b k x^ converges everywhere, then our theorem evidently holds for every x for which 2a n x n converges absolutely. 88 We have therefore to write a^ 0) = 1 , aj * = a.J 0) = = , and the latter for n = 0, 1 , 2 , . . .. 7 182 Chapter V. Power series. If both series converge everywhere, then the theorem holds without restriction for every ar. 2. If a = and both series have a positive radius, then the theorem ceitainly holds for every ''sufficiently" small a, that is to say, there is then certainly a positive number g, such that the theorem holds for every | a; | < For if y = a t x -f <z a x* -f- i then rj = | a \ \ x \ -f- \ a 2 \ \ x* \ + ; and since for x > , we now also, by 96, have ij > 0, rj is certainly less than the radius of 2b k y k for all x whose absolute value is less than a suitable number g. y n 3. In the series JE7 - , we "may" for instance substitute y = 2 x n for x n | x | <C 1 i or y for every n , and then rearrange in powers of x . 4. To write, as we did above: oi2 is, we now see, certainly allowed if ={= and further x is in absolute value so small that - x fl a n which by Rem. 2 is certainly the case for every | x \ <[ Q with a suitable choice of Q . We may therefore say: We "may" divide by a power series of positive radius if ^ts constant term is =(= and provided we restrict ourselves to sufficiently small 20 values of x. To determine the coefficients c n by the general method used to prove their existence, would, even for the first few indices, be an extremely laborious process. But once we have established the possibility of the expansion which is at the same time necessarily unique by 97, we may determine the c n 's more rapidly by remarking that 2a n xP'2c n gpss l f so that we have successively - From these relations, since a Q ^ 0, the successive coefficients c 0t c^ c z , . . . may be uniquely determined, the simplest method being with the aid of determinants, by Cramer's Rule, which immediately yields a closed expression 3o for c n in terms of a , a l9 . . . , a n . 5. As a particularly important example for many subsequent investigations we may set the following question 31 : ^ 29 How small x has to be, is usually immaterial. But what is essential, is that some positive radius g exists, such that the relation holds for every | x \ < Q. The determination of the precise region of validity requires deeper methods of function theory. 80 Explicit formulae for the coefficients of the expansion, in the case of the quotient of two power series, may be found e. g. in y. Hagen t On division of series, Americ. Journ. of Math., Vol. 5, p. 236, 1883. 81 Euler: Institutions calc. diff., Vol. 2, 122. 1755. 21. The algebra ot power series. 183 in powers of x. Here the determination of the new coefficients becomes peculiarly elegant if we denote them, not by c nt but by ~, or as we shall ~D do, for historic reasons, by ~. Then the above equation is and the equations for determining B n are, in succession, B -1 1B ' + l.*i-0 ^0-1. 2,01+H 1!~ U ' and, in general, for n = 2, 3, . . . , .__.__ . .. !"0'" t "(n-l)l'l!" r (n-2)l 2I" 1 " " " r II (n - 1)1 If we multiply by n\ y we may write this more concisely: Now if we here had r in place of B v , for each v, then we could write instead w JB n =0; 106. and the recurring formula under consideration also may bo borne in mind under this convenient form, as a symbolic equation, i. e. one which is not intended to be interpreted literally, but only becomes valid with a particular convention, here the convention that after expanding the w tb power of the binomial (#+1), we replace each B v by B v . Our formula now yields, for n = 2, 3, 4, 5,.. successively, the equations 2/^ + 1=0, 3 B 2 + 3 B! + 1 = , 5 B 4 + 10 a, + 10 # 2 + 5 B, + 1 = , from which we deduce and then and * 14 = 6 These are called Bernoulli's numbers and will be mentioned repeatedly later on ( 24,4; 32,4; 55, IV; 64). For the moment, we are able to infer only that the numbers B n are definite rational numbers. They do not, however, conform to any apparent or superficial law, and have formed the subject of many elaborate discussions 88 . 32 Bernoulli's numbers are frequently indexed somewhat differently, B Qt B lt BQ B 6 ,B 7I ... being omitted and ( l)*" 1 ^ written instead of J5 flfc , for k = 1, 2, ... A table of the numbers J9 a ,l? 4 , ..., to B m may be found in /. C. Adams, Journ. f. d. leine u. angcw. Math., Vol. 85, 1878 We may mention in passing that U wo ha^ for numerator a number with 113 digits, and for de- nominator the number 2 358 255 930; while # 123 has the denominator 6 and, 184 Chapter V. Power series. Finally we will prove one more general theorem on power series: CO Given the power series y = 2 a n (x x ) n , convergent for \ x x \ <r, w-O we have, for every x in the neighbourhood of X Q , a determinate corre- sponding value of y, in particular for x =-= x the value y = a Qt which we will accordingly denote by y . Then we have y - y = i (x - * ) + 2 (* - *o) 2 + - Because of the continuity of the function, to every x near x also corre- sponds a value of y near j> . We would now enquire whether or how far every value of y near y is obtained and whether it is obtained once only. If the latter was the case, not merely y would be determined by x, but con- versely x would be determined by y, and therefore x would be a function of y. The given function y=f(x) would, as we say for brevity, be reversible in the neighbourhood of X Q (cf. 19, Theorem 6). The question of reversibility is dealt with by: 107. Reversion theorem for power series. Given the expansion y - :xo = i (* - * ) 4- a (* - *o) 2 + > convergent for \ x x \ <r, the function y =f(x) thereby determined is reversible in the neighbourhood of X Q> under the sole hypothesis that a 4= U; i. e. there then exists one and only one function x 9 (y) which is expressible by a power series, convergent in a certain neighbourhood of y , of the form * - *o = *i (y - y ) + b 2 (y - j ) 2 + . . . and for which, in that neighbourhood, we have (in the sense of 104) Moreover b 1 = 1 : a v Proof. As we have already done more than once, we assume in the proof that x and y are =0, which implies no restriction 33 . But we will then further assume that a l = 1, so that the expansion (a) y = x + * 2 x * + *3 * 3 + is the one to be reversed. That too implies no restriction, for since a 1 4= 0, by hypothesis, we can write a^ x + a 2 x 2 + . . . in the form in the numerator, a number with 107 digits. The numbers B z , B 4t . . . , to B 6Z had previously been calculated by Ohm, ibid., Vol. 20, p. Ill, 1840. The numbers B v first occur in James Bernoulli, Ars conjectandi, 1713, p. 96. A com- prehensive account is given by L. Saalschutz, "Vorlesungen uber die Bernoulhsclnen Zahlen", Berlin (J. Springer) 1893, and by AT. E. Norlund, "Vorlesungen uber Differenzenrechnung", Berlin (J. Springer) 1924. New investigations, which chiefly concern the arithmetical part of the theory, are given by G. Frobenius, Sitzgsber. d. Berl. Ak., 1910, p. 809847. 33 Or: we write for brevity x X Q = x' and y y Q = y' and then, for sim- plicity's sake, omit the accents. 21. The algebra of power series. 185 If we write for brevity a x = a/ and, for n ^> 2, and subsequently, for simplicity's sake, omit the accents, then we obtain precisely the above form of expansion. It suffices therefore to consider this. But we can then show that a power series, convergent in a certain interval, of the form (b) x = y + \ y 2 + b 3 y 3 -\ exists which represents the inverse function of the former, so that is identically = y, if this series is arranged in powers of y, in accor- dance with 104, i. e. all the coefficients must be = except that of y 1 , which is = 1. Since we have written, for brevity, x instead of a 1 x, we see that the series on the right hand side of (b) has still to be divided by <z to represent the inverse of the series a t x -f- <z a x* -j- , where a t has no specialised value. In this general case we shall therefore have b 1 = as coefficient of y 1 . If we assume, provisionally, that the statement (b) is correct, then the coefficients b v are quite uniquely determined by the condition that the coefficients of y 2 , y*, ... in (c) after the rearrangement, have all to be =0. In fact, this stipulation gives the equations , + a* = 64 from which, as is immediately evident, the coefficients b v may be determined in succession, without any ambiguity. Thus we obtain, the values but the calculation soon becomes too complicated to convey any clear idea of the whole. Nevertheless, the equations we have written down show that if there exists at all an inverse function of y = f(x), capable of expansion in form of a power series, then there exists only one. Now the calculation just indicated shows that whatever may have been the original given series (a), we can invariably obtain perfectly 186 Chapter V. Power series. determinate values & r , so that we can invariably construct a powei series y -\- & 3 y 9 -|- which at least formally satisfies the conditions of the problem, the series (c) becoming identically = y. It only remains to be seen whether the power series has a positive radius of convergence. If that can be proved, then the reversion is completely carried out. The required verification may, as Cauchy first showed, actually be attained, in the general case, as follows: Choose any positive numbers cc for which we have and 2 cc v x v has a positive radius of convergence. Proceeding in the above manner, for the series: y = x a 2 x* whose inverse is, then, say, we obtain, for the coefficients /? y , the equations ft = GV + 2 Ai) 8 3 + 3 A s 4- ....... ........... in which all the terms are now positive. Thus for every r. If, therefore, it is possible so to choose the cc v that the series 2 ft v y v has a positive radius of convergence, it would follow that 2 b v y v also had a positive radius and our proof would be complete. We choose the ccjs as follows: There is certainly a positive number Q, for which the original series x -f- a x* -\- converges absolutely. A positive number K must, however, then exist (by 82, Theorem 1 and 10, 11) such that we have, for every r = 2, 3, ..., We then choose, for y = 2, 3, . . ., so that we are concerned with reversing the series, convergent for \*\<e> But this function is immediately reversible. For we may at once see by differentiation we are dealing, in fact, with a simple hyperbola, of which the student should draw a graph for himself , that in 00 < X < X l 21. The algebra of power series. 187 the function increases monotonely (in the stricter sense) from oo to the value and therefore possesses, for y < y lf a uniquely determined inverse whose values arc < a? 1 . For this, since y = x ~~ efe'-^g) r (X + e ^ 9 ~" e k + y }* + Q * y = ' we have, uniquely, Further if we write for brevity, with the above defined value of y t , and both y l and y 2 are >> 0, since the second is and the two have product = Q 2 . But In the following chapter we shall see that, for 1 2 1 < 1, the power (1 2)! can actually be expanded in a power series beginning with 1 -*- + ' ' Assuming this result, it follows immediately that & x also may be expanded in a power series, convergent at least for X = By our first remarks the proof is hereby entirely completed. The actual construction of the series y + & 2 y 9 + from the series * + a ** H here also involves in general considerable difficulties and necessitates the use of special artifices in each particular case 34 . Examples of this will occur in 26, 27. We only note further, a fact which will be of use later on, that if (b) is the inverse of (a), then the inverse of the series (a') y = x-a^-\-a^ - H where the signs are alternated, is obtained from (b) by similarly alternating the signs, i. e. (V) * = y - b* y* + b a y 3 - + 34 The general values of the coefficients of expansion b n are worked out as far as 6 13 by C. E. van Orstrand, Reversion of power series, Philos. Magazine (6), Vol. 19, p. 366, 1910. 188 Chapter V. Power series. This is at once evident, if we first actually expand the powers of in (c), obtaining, say, (c) (y Under the new assumption, the same process, since the product of two series with alternating coefficients is again a series with alternating co- efficients, gives (c') (y - brf + ...)-. (J 2 - * (8 >^ + - - ) And from this we immediately infer that on equating to zero the coeffi- cients of y 2 , y 3 , . . . , we must obtain the identical equations (d), thus de- ducing for b v precisely the same values as before. The exact analogue holds good when the two power series contain, from the first, only odd powers of x. Thus, if the inverse series of y = X -f tf 3 #3 + a 5 X 5 -f- . .. is x = y + b 3 y* + b 5 y 5 + . . . , then the inverse series of y = xa 3 afi + az3f-\-... is necessarily x = y b. 3 y 3 + b 5 y 5 -- 1- . . . . Exercises on Chapter V. 64. Determine the radius of convergence of the power series S a n x n t when a n has, from some point onwards, the values given in Ex. 34 or 45. 65. Determine the radii of the power series 0<*<1; 66. Denoting by y. and p, the lower and upper limits of , the radius r of the power series Z a n x n invariably satisfies the relation x ^ r ^ p. In par- ticular: If lim exists, it has for value the radius of S a n x n . \ a n+l 67. Sa n x n has radius r, Z a n ' x n radius r'. What may be said of the radius of the power series a n n 67 a. What is the radius of Z a n x n if < hm | a n \ < + oo? 00 68. The power series -, n x n t where e n has the same value as in Ex. 47, n converges at both ends of the interval of convergence, but in either case only con- ditionally. 69. Prove, with reference to 97, example 3, that s ("y-(-i)l(-ir( 2 y- ft"). K=O W ^o \ v / \ " / 22. The rational functions. 189 70. As a complement to Abel's theorem 100, it may be shewn that in every case in which E a n x n has a radius r ^ 1, we have (00 \ _ 2 <*n x ) ^ lim* n *-^*-- v n = Q I (s n o + i 4 + tf w )- 71. The converse of AbeVs theorem 100, not in general true, holds, however, if the coefficients a n are ^ ; if therefore, in that case, hm 2a n x n *->r-0 exists, then 2 a n r n converges and its sum is equal to that limit. 72. Let % a n x n - / (x) and Z b n x n - g (*), 71-1 H=l both series converging for | x \ < Q. We then have (for what values of #?) n-l M-=! (By specialising the coefficients many interesting identities may be obtained. Write e.g.i n = !,(-!)-', ~, etc.) 73. What are the first terms of the series, obtained by division, for (Further exercises on power series will be found in the following Chapter.) Chapter VI. The expansions of the so-called elementary functions. The theorems of the two preceding sections ( 20, 21) afford us the means of mastering completely a large number of series. We proceed to explain this in the most important cases. A certain not very large number of power series, or functions represented thereby, have a considerable bearing on the whole of Analysis and are therefore frequently referred to as the elementary functions. These will occupy us first of all. 22. The rational functions. From the geometric series ! + * + * + . ..= *= * |*| <1, w =o L ~ x which forms the groundwork for many of the following special investi- gations, we deduce, by repeated differentiation, in accordance with 98, 4: !<" and generally, for any positive p : 1. 1< 1.108. (G51) 190 Chapter VI. The expansions of the so-called elementary functions. If we multiply this equation once more, in accordance with 91, by 2x n =, we obtain, by 01 and 108: By comparing coefficients (in accordance with 07), we deduce from this that P J ' ' V P which may of course be proved quite easily directly (by induction). If we do this, we may also deduce the equality 108 by repeated mul- tiplication of x " i w * m itself, by 103. 1 x Since we have (n + p\ fn-rp\ , -t\nfp ^\ ( P ) = ( n j = (- 1 ) ( n ) we obtain from 108, if we there write x for x and k for p +1> the formula 109. valid for |a?| <1 and negative integral k. This formula is evidently an extension of the binomial theorem (29, 4) to negative integral exponents; for this theorem may for positive integral k (or for k = 0), also be written in the form 109, as the terms of the series for n > k are in that case all =0. Formulae such as those we have just deduced have as we may observe immediately, and once for all a two -fold meaning; if we read them from left to right, they give the expansion or representation of a function by a power series; if we read them from right to left, they give us a closed ex- pression for the sum of an infinite series. According to circumstances, the one interpretation or the other may occupy the foremost place in our attention. By means of these simple formulae we may often succeed in expanding, in a power scries, an arbitrary given rational function namely whenever f(x) may be split up into partial fractions, i. e. ex- pressed as a sum of fractions of the form A (x-a)' Every separate fraction of this kind, and therefore the given func- tion also, can be expanded in a power series by 108. And in fact this expansion can be carried out for the neighbourhood of every point x distinct from a. We only have to write ( )' \x-aj /*- U -* 23. The exponential function. 191 and then expand the last fraction by 108. By this means we sec, at the same time, that the expansion will converge for | x x \ < | a x \ and only for these values of x. This method, however, only assumes fundamental importance when we come to use complex numbers. Examples. ^ HO. 1 V - 2 2V* ?-I*~-_ - 4 * ^J C\n & " ^j t M * C 3. ' 23. The exponential function. 1. Besides the geometric series, the so called exponential series GO ~n .'J r 3 /v.n 2 - = i + x + - -\- - H h ^r H n=0 n ' .o n. plays a specially fundamental part in the sequel. We proceed now to examine in more detail the function which it represents. This so-called exponential function we denote provisionally by E(x). As the seiies converges everywhere by 92,2, E (x) is certainly, by 98, defined, continuous and differentiate any number of times, for every x. For its derived function, we at once find so that for all derived functions of higher order we must also have We shall attempt to deduce all further properties from the series itself. We have already shown in 91, 3 that if x and x are any two real numbers, we have in all cases (a) (XL + aco) = E (x\)-E (x>>) . This fundamental formula is referred to briefly as the addition theorem for the exponential function 1 . It gives further and by repetition of this process, we find that for any number of real numbers x l9 # a , . . ., x k , 1 Alternative proof. The Taylor's series 99 for E(x) is valid for all values of x and x . If we observe that E^ fa) = E fa), then it at once follows, replacing x by a^+tfg, that q. e. d. 192 Chapter VI. The expansions of the so-called elementary functions. If we here write x = 1 for each r, we deduce in particular that holds for every positive integer k. Since E (0) = 1, it also holds for k = 0. If we now write, in (b), x v = ~ for each y, denoting by m a secon^ integer ^ 0, then it follows that or, ~ since (w) = [(!)] w , that If we write for brevity E(i) = E, we have thus shewn that the equation (c) E(x) = E x holds for every rational x^>0. If is any positive irrational number, then we can in any number of ways form a sequence (x n ), of positive rational terms, con- verging to f . For each n, we have, by the above, When w *-f co, the left hand side, by 98,2, tends to(f), and the right hand side, by 42, 1, to E , so that we obtain E(f) = E . Thus equation (c) is proved for every real x I> 0. But, finally, (a) gives E(- *)-(*) = E(a - a) = E(0) = 1, whence we first conclude that E (x) = cannot hold 2 for any real x and that for a; . E(x) E x But this implies that equation (c) is also valid for every negative real x. We have thus proved that the equation holds for every real x\ and at the same time the function E(x) has justified its designation of exponential function; E (x) is the ar th power of a fixed base, namely of l + i + i + ^ + ... + i + ... 2 This may of course, for x > , be deduced immediately from the series, by inspection, since this is a series of positive terms whose term of rank is = 1. 23. The exponential function. 193 2. It will next be required to obtain some further information about this base. We shall show that it is identical with the number e already met in 46 a, so that 3 Mm The proof may be made somewhat more comprehensive, by at once establishing the following theorem, and thus completing the investi- gation of 46, a: o Theorem. For every real x 9 111. / x \ x v lim (1 -| -- ) exists and is equal to the sum 4 of the series Z -,. n ->x w ' " v -i)V\ Proof. We write for brevity (* +)"=* and !;$=(*)= It then suffices to prove that (s x n ) * . Now if, given, first, a definite value for x, e is chosen > 0, we can assume p so large that the remainder 2 ' Further, for n > 2, a series which terminates of itself at the n ih term. The term in x k , k 0, 1, . . . , evidently has a coefficient ^> 0, but not greater than the coefficient 1/&I of the corresponding term of the exponential series. The same is also true, therefore, of the difference of the former and the latter term. Accordingly we have, for n > p from the manner in which p was chosen 5 Every individual term of the (p 1) first terms on the right hand side 8 We have here, therefore, a significant example of problem B. Cf. intro- duction to 9. 4 First proved if not in an entirely irreproachable manner by Enter, Introductio in analysin infinitorum, Lausanne 1748, p. 86. The exponential series and its sum e x were already known to Newton (1669) and Leibniz (1676). 6 We assume p>2 from the first. 194 Chapter VI. The expansions of the so-called elementary functions. is now obviously the n th term of a null sequence"; hence their sum for p is a fixed number also tends to 0, and we may choose w > p so large that this sum remains < ~ for every n > n {} . But we then have, for every n > n , \s-x n <6, which proves our statement 7 . For x = 1 , we deduce in particular 00 I / = ^-l=lim(l + - y=0 vl r-> oo ^ and more generally, for every real x y The new representation thus obtained for the number e, by the exponential series, is a very much more convenient one for the further discussion of this number. In the first place, we can, by this means, easily obtain a good approximation to e. For, since all the terms of the series are positive, we evidently have, for every n, or . e. 8 We have (l_JL)->l f (i-jL\-+i 9 ..., fl -^-1)-* 1, and so their product (by 41, 10), also -> 1, or 1 - f 1 J . . . (l ^"" J -^0; so, as x and ^ are fixed numbers, the product of this last expression by T|#|^ also -*0; and similarly for the other terms. We can also infer the result directly from 41, 12 7 The artifice here adopted is not one imagined ad hoc, but one which is frequently used: The terms of a sequence are represented as a sum x n = a; (n) + x^ + + fcj. n \ where the terms summed not only depend in- dividually on n, but also increase in number with n: k n > OO . If we know how each individual term behaves for n > oo , as for instance, that x v (n) for fixed v tends to v , then we may often attain our end by separating 1 out a fixed number of terms, say o: (n) -\-x^ (n) -f- -\-x p ^ with fixed p\ this tends, when n -> oo , to 4- f l -f + f , by 41, 9. The remaining terms, *> + . . . -f *> we then endeavour to estimate in the bulk directly, by finding bounds above and below for them, which often presents no difficulties, provided p was suitably chosen. 23. The exponential function. 195 \vherc s n denotes a partial sum of the new series for e. If we cal- culate these simple values e. g. for n = 9 (v. p. 251) then we find 2 - 718 281< e < 2 718 282 , which already gives us a good idea of the value 8 of the number e. From the formula (a) we may, however, draw further important infeiences. A number is not completely before us unless it is rational and is written in the form . Is e perhaps a rational number? The inequalities (a) show quite easily that this is unfortunately not the case. For if we had e = , then for n = q, formula (aj would give: where s q = 2 -f- ^ -| ---- -f r. If we multiply this inequality by q\, then q \ s q is an integer, which we will denote for the moment by g, and it follows that But this is impossible; for between the two consecutive integers g and g -f- 1 there cannot be another integer p - (q l) 1 distinct from either : e is an irrational number. 3. The above investigations give us all the information, with regard (x \ n 1 -| -- ) , which we, in the first instance, require; the two problems A and B ( 9) are both satisfactorily solved. In spite of this, we propose, in view of the fundamental importance of these matters, to determine the same limit again and in a different way, entirely independent of the preceding. (1 \ n 1 -| -- ) > e . This we will first extend by showing that * n alsoy when (yj is any sequence of positive numbers tending to + oc- When y n = a positive integer, for every n, this is an immediate con- sequence of the previous result 9 . 8 The number e nas been calculated to 346 places of decimals by J. M. Boormann (Math, magazine, Vol. 1, No. 12, p. 204, 1884). 9 For if e is given > , and n is determined, by 46 a, so that remains .< * for every n>n , then we shall also have (1 \ v I 1 _] --- J n e\ <^e for every n > tt lf provided n v is so chosen that for every y/ I n >> n 4 we have y n ;> . M -| --- J _ 196 Chapter VI. The expansions of the so-called elementary functions. If the numbers y n are not integers, there will still be for each n one (and only one) integer k n such that *.y.<*. + l>. and the sequence of these integers k n must evidently also tend to -|- oo. Now, however, if k n ^ 1, And since the numbers k are integers, the sequence (>+ and the sequence both tend to e, by our first remark. Hence, by 41, 8, we also have (+f>- We may next show that when y n '+ oo, we also have or, otherwise, that when y n > -|- oo, we have / 1 \-Vn 1 -- ) -+e. \ yJ All the numbers y n ' must, however, be assumed < 1, i. e. y n > 1 , so that the base of the power does not reduce to or a negative value; this can always be brought about by "a finite number of alterations". Since and since, with y n , y n 1 also > -|- oo, the statement to be proved is an immediate consequence of the preceding one. Writing = z n , we may couple the two results thus: provided (z n ) is any null sequence with only positive or only negative terms, the terms in the latter case being all > 1. From this we finally obtain the theorem, including all the above results: Theorem: // (xj is an arbitrary null sequence whose terms are different from and > 1 from the first 10 , then u i (a) Urn (1 + *,)* . 10 The latter may always be effected by "a finite number of alterations' (cf. 38, 6). 11 Cauchy: Rgsume' des lemons sur le calcul infinit., Paris 1828, p. 81. 23. The exponential function. 197 Proof. Since all the x n 's =J= 0, the sequence (# n ) may be divided into two sub-sequences, one with only positive and one with only ne- gative terms. Since, for both sub-sequences, the limit in question, as we have proved, exists 12 and = e, it follows by 41, 5 that the given sequence also converges, with limit e. By 42, 2, the result thus obtained may also be expressed in the form (b) log, (1 + **) _ ! y x n which will frequently be used. By 19, Def. 4, the result also signifies that, invariably: From these results, it again follows, quite independently, as we announced, of our investigations of 1. and 2. , that ('+3'-" for ( J is certainly a null sequence 18 , so that we have, by the pre- ceding theorem, n e and therefore (l + -)"-* e 9 , which was what we required 14 . 4. If a > 0, and x is an arbitrary real number, then, denoting by log the natural logarithm (v. p. 211), a x = gxioga = 1-1- 9* x + x 2 4- S* x? 4- C* G X T j i * ~ 2 I I Q I ** I is an expansion in power series of an arbitrary power. We deduce the limiting relation 15 2Lni->loga for *->0, a> 0.113 18 If one of the two sub-series breaks off after a finite number of terms, then we can, by a finite number of alterations, leave it out of account. 18 We consider this null sequence for n > | x \ only, so that we may al- x ^ t ways have > 1 . 14 Combining this with the result deduced in 2., that the above limit has the same value as the sum of the exponential series, we have a second proof of the fact that the sum of the exponential series is =e*. 16 Direct proof: If the x n 's form a null sequence, then by 35, 3, so do the numbers ? = ** 1; and consequently, by 112 (b), a x *l y*log-a log- a _->_- = log a. 198 Chapter VI. The expansions of the so-called elementary functions. This formula provides us with a first means of calculating loga- rithms, which is already to a certain extent practicable. For it gives, e. g. (cf. 9, p. 78) log a = lim n (ya 1^ n->o> 2 *__ =lim2*(Va-l). *->oo As roots whose exponent is a power of 2 can be calculated directly by repeated taking of square roots, we have in this a means (though still a primitive one) for the evaluation of logarithms. 5. We have already noted that e* is everywhere continuous and differentiable up to any order, with e x = (e*}' = (e*)" = . It also shares with the general power a x ', of base a > 1, the property of being everywhere positive and monotone increasing with x. More noteworthy than these are the properties expressed by a scries of simple inequalities, of which we shall make use repeatedly in the sequel, and which are mostly obtained by comparison of the exponential with the geometric series. The proofs we will leave to the reader. 114. a) For every 16 x, e? > 1 + x, ft) for x<l, <**<, for x> -1, 8) for x < + 1, x < e* - 1< j^, X e) for x > 1 , l-\-x> e^~ x , rn f) for x > 0, c*>-p\> (P = > 1. 2, . . .), rj) for x > and y > 0, ** > (l + ^ > *^+5, for every z + 0, |* l| < * M K 24. The trigonometrical functions. We are now in a position to introduce the circular functions rigorously, i. e. employing purely arithmetical methods. For this pur- pose, we consider the series, everywhere convergent by 92, 2: 16 Only for x = do these and the following inequalities reduce to equa- lities. The reader should illustrate the meaning of the inequalities on thfe relative curves. 24. The trigonometrical functions. 199 and , a* a* . i / iV S(a) - x - - 3l + 5f - H ----- |- (- 1) pTj-.pl)] -- Each of these series represents a function everywhere continuous and differentiable any number of times in succession. The properties of these functions will be established, taking as starting point their ex- pansions in scries form, and it will be seen finally that they coincide with the functions cos a; and sin a; with which we are familiar from elementary studies. 1. We first find, by 98, 3, that their derived functions have the following values: r9 _ c- f*u _ r* rill _ r xv/// _ C \_s ~~~ O , O ~~~~ v^ j ^ tJ ) \^f O , Qf _ f+ C" _ C C'" _ /"* o//// _ c . J -- O y O ~~~* O , O ~~~~ \s , O O , relations valid for every x (which symbol is for brevity omitted). Since, here, the 4 th derived functions are seen to coincide with the original functions, the same series of values repeats itself, in the same order, from that point onwards in the succession of differentiations. Further, we see at once that C(x) is an even, and (#) an odd, function: , N >-/\ o/ \ o/\ C ( x) = C (x) , 5 ( x) = 5 (x) . These functions also, like the exponential function, satisfy simple ad dition theorems, by means of which they can then be further examined. They are most easily obtained by Taylor 's expansion (cf. p. 191, foot- note 1). This gives, for any two values x t and a? a , since the two series converge everywhere (absolutely), and as this series converges absolutely, we may, by 89, 4, rearrange it in any order we please, in particular we may group together all those terms for which the derived functions which they contain have the same value. This gives -- Tr (a) C (x, + ar a ) == C (*J C (x,) - S (zj S (*,) ; and we find 17 quite similarly (b) S fo + ,) = S (*,) C (*,) + C ( 17 Second proof. By multiplying out and rearranging in series form, we obtain from C(x t )C(x a )-S(x l )S(x a ) the series C (x l + x^) , as in 91, 3 for the exponential series. Third proof. The derived function of f(x) = [C (x, + x)-C (x,) C (x) + S (X L ) S (x)]* + [S (x, + x)-S (xj C (x) - C (xj S (*)f is, as may at once be seen, =0. Consequently (by 19, theorem 7), f(x)^f(Q) = Q. Hence each of the square brackets must be separately ~ 0, which at once gives both the addition theorems. 200 Chapter VI. The expansions of the so-called elementary functions. From these theorems, whose form coincides with that of the addition theorems, with which we are already acquainted from an ele- mentary standpoint, for the functions cos and sin, it easily follows that our functions C and 5 also satisfy all the other so called purely goniometrical formulae. We note, in particular: From (a), writing x^ = x 9 we deduce that, for every x, (c) C(*) + S(*) = 1; from (a) and (b), replacing both x and x% by x: S(2x)=2C(x)S(x). 2. It is a little more troublesome to infer the properties of periodicity directly from the series. This may be done as follows: We have C (0) = 1 > . On the other hand, C(2) < 0; for C(2^-l---i- 24 -^ ^)/^o 2 ^\*) L 21 ' 41 \6! Si) \10! I2l) "" where the expressions in brackets are all positive, since for nl>2, + _ ' n! A. 1 A t and therefore C (2) < 1 -^ + 4 ~ ~~ "3" ' '* e ' certainl y negative. By 19, Theorem 4, the function C (x) therefore vanishes at least once between and 2. Since further, as may be again easily verified, is positive for all values of x between and 2, and therefore C'(x)= S(x) constantly negative there, it follows that C (x) is (strictly) monotone decreasing in this interval and can only vanish at one single point f in that interval. The least positive zero of C(x), i. e. f, is accordingly a well defined real number. We shall imme- diately see that it is equal to a quarter of the perimeter of a circle of radius 1 and we accordingly at once denote it 18 by : From (c), it then follows that S 2 f~J = l, i. e. since S (x) was seen to be positive between and 2, that = 1. 18 The situation is thus that n is to stand for the moment as a mere ab- breviation for 2; only subsequently shall we show that this number n has the familiar meaning for the circle. 24. The trigonometrical functions. The formulae (d) show further that and by a second application, that 201 It then finally follows from the addition theorems that, for every x, c(x + ?) = -S(x), \ z / /~ / i \ /^ / M \ U (X -f- 71) = C (CC) , \ ' / \ / ' C(TT~- a;)= C(*), Our two functions thus possess 19 the period 2 TT. 3. It therefore only remains to show that the number n, intro- duced by us in a purely arithmetical way, has the familiar geometrical significance for the circle. Thereby we shall have also established the complete identity of our functions C (x) and S(x) with the functions cos re and sin a; respectively. Let a point P (fig. 3) of the plane of a rectangular coordinate system OXY, be assumed to move in such a manner that, at the time t> its two coordinates are given by and then its distance | O P \ = Vrc 9 -f- y* from the origin of coordinates is constantly = 1, by (c). The point P therefore moves along the peri- meter of a circle of radius 1 and centre O. If, in particular, t increases from to 2 n, then the point P starts from the point A of the positive #-axis and describes the peri- meter of the circle exactly once, in the mathe- matically positive (i. e. anticlockwise) sense. In fact, as / increases from to n, x = C (t) decreases, as is now evident, from -f- 1 to 1, monotonely, and the abscissa of P thus assumes each of the values between -f- 1 and 1, exactly once. At the same time, S(f) remains constantly positive; this therefore implies that P describes the upper half of the circle from A to B steadily, and passes through 19 2 n is also a so-called primitive period of our functions, i. e. a period, no (proper) fraction of which is itself a period. For the formulae (e) show that -^ = n is certainly not a period. And a fraction , with m>>2, cannot be 2i nt (O - \ \ = S(0) = 0, which is impossible since S(a;) was seen to be positive between and 2 and in fact, as S (n x) = S (x) , is positive 2 31 between and n. Similarly for C (a;), (m > 1) cannot be a period. Fig. 3. 202 Chapter VI. The expansions of the so-called elementary functions. each of its points exactly once. The formulae (e) then show further that when t increases from n to 2 n, the lower semi-circle is described in exactly the same way from B to A . These considerations provide us first with the Theorem. // x and y are any two real numbers for which x*-\-y 2 = I, then there exists one and only one number t between (incl.) and 2n (excl.\ for which, simultaneously, C(t) = x and S (t) = y . If we next require the length of the path described by P when t has increased from to a value t Qy the formula of 19, Theorem 29 gives at once, for this, the value In particular, the complete perimeter of the circle is 2n ___ SJT == / Vt' 2 + ST*dt = / dt = 2 n . The connection which we had in view between our original conside- rations and the geometry of the circle, is thus completely established: C (t), as abscissa of the point P for which the arc A P = t, coincides with the cosine of that arc, or of the corresponding angle at the centre, and S (t), as ordinate of P, coincides with the sine of that angle. From now on we may therefore write cos t for C(f) and sintf for S(t). Our mode of treatment differs from the elementary one chiefly in that the latter introduces the two functions from geometrical considerations, making use naively, as we might say, of measurements of length, angle, arc and area, and from this the expansion of the functions in power series is only reached as the ultimate result. We, on the contrary, started from these series, examined the functions defined by them, and finally established using a concept of length elucidated by the in- tegral calculus the familiar interpretation in terms of the circle. 4. The functions cota; and tana; are defined as usual by the ratios cos x sin re cota; = , tan x = -- : sin x cos x as functions, they therefore represent nothing essentially new. The expansions in power series for these functions are however not so simple. A few of the coefficients of the expansions could of course easily be obtained by the process of division described in 105, 4. But this gives us no insight into any relationships. We proceed as follows: In 1O5, 5, we became acquainted with the expansion 20 20 The expression on the left hand side is defined in a neighbourhood of exclusive of this point; the right hand side is also defined in such a neigh- bourhood, but inclusive of 0, and moreover is continuous for x~ 0. In such case we usually make no special mention of the fact that we define the left hand side for x = by the value of the right hand side at the point. 24. The trigonometrical functions. 203 1 v =o "1 2 where the Bernoulli's numbers B v are, it is true, not explicitly known, but still are easily obtainable by the very lucid recurrence formula 100. These numbers we may, and accordingly will, in future, regard as entirely known 21 . We have therefore, for every "sufficiently" small x (cf. 1O5, 2, 4) x * 1 [ B * x * I ... ^i^a""" 1 ^ 21 ^ ' The function on the left hand side is however equal to jL _-?. _ x / 2 \ _ xe *+\ _xe 2 +e 2 """ 'A** 1 ' / ~~ 2 *-! ~~ 2 *. _* e*-e * and from this we see that it is an even function. Bernoulli's numbers J5 3 , B$ y B 7 are therefore, by 97, 4, all = 0, as already seen in 106, and we - x have, using the exponential series for e* and writing for brevity z: ST 5T If on the left hand side, we had the signs -j- and occurring alter- nately, both in the numerator and denominator, we should have pre- cisely the function zcotz. Dividing out on the left hand side by the factor z y so that only even powers of z occur, may we then deduce straight away that the relation , co t,_ obtained from our equality by alternating the signs throughout, is also valid? Clearly we may. For if, to take the general case, we have for every sufficiently small z: the same relation holds good when the + signs throughout are re- placed by alternate + and signs. In either case, in fact, the coeffi- cients c 2v are obtained, according to 105, 4, from the equations: c* + *>9 *=**'> c * + C a 6 2 +*>4 = &4 9 > ^ + c 4 6. J + c a 6 4 +6 tf =a a ; ...; czv + c 2 ,-2 & a H ----- h c a &Jv~2 + b-2 V = a 2v ; ... fl As appears from the definition, they are certainly all rational. 204 Chapter VI. The expansions of the so-called elementary functions. We therefore, as presumed, now writing x for z, have the formula 22 : ^ ft __1 l 9 1 4 2 l 8 l ~~~S X ""45* ~~945* ""4725* The expansion for tan x is now most simply obtained by means of the addition theorem ft rt cos 2 x sin 2 # 2 cot 2 a; = : ~ cot x tana; f cosar-smo; f from which we deduce tan x = cot x 2 cot 2 3 and therefore 23 * / v ^x 1 2 8 *(2 s *-l)J?, Jfc o ^ 1 16. (a) tan x = ( l) k ~ l ^-^j M* h ~ l = x-\ --a; 8 -I 2 g g -l 1? x 7 From the two expansions, with the help of the formula 1 cot + tan -^ = -7 ' 2 sin # we obtain further tina? . I X n I I 4 I A I 4]| a: +'" (An expansion for 1 /cos re will be found on p. 239.) These ex- pansions, at the present point, are still unsatisfactory, as their interval of validity cannot be assigned; we only know that the series have a positive radius of convergence, not, however, what its value is. 5. From another quite different starting point, Euler obtained an interesting expansion for the cotangent which we proceed to deduce, especially as it is of great importance for many problems in series 34 . At the same time, it will give us the radius of convergence of the series 115 and 116 (v. 241). * 2 This and the following expansions are almost all due to Euler and are found in the 9 th and 10 th chapters of his Introductio in analysin infimtorum, Lausanne 1748. 28 We shall afterwards see that B 2k has the sign (-1)*" 1 (v. 136), so that the expansion of zcotg, after the initial term 1, has only negative coeffi- x cients, those of tanx and only positive coefficients. M The following considerable simplification of Euler' s method for obtaining the expansion is due to Schrdter (Ableitung der Partialbruch- und Produkt- entwicklung-en ttir die trigfonometrischen Funktionen. Zeitschrift fUr Math. u. Phys., Vol. 13, p. 254. 1868). Z*. me trigonometrical lunctions. 205 We have, as was just shewn, or a formula in which we may, on the right, take either of the signs . Let x be an arbitrary real number distinct from 0, 1, 2, ..., whose value will remain fixed in what follows. Then jt X ( t 3i x , yi (x -f- 1) } jixcotnx = - 2 - 1 cot ~p- -f- cot 21 and applying the formula (*) once more to both functions on the right hand side, taking for the first the -f- and for the second, the sign, we obtain jix f nx , f 3t(x + l) , jc(x-l)l . ^(# + 2)1 n x cot n x = -j- |cot - f - + [cot - A -~ -f cot v 4 ; J -f cot ~~-^ j . A third similar step gives, for jixcotjtx, the value + cot + cot + cot + cot m^> + cot ^L^L + cot H* 8 =^ since here each pair of terms which occupy symmetrical positions relatively to the centre () of the aggregate in the curly brackets give, except for a factor 1, a term of the preceding aggregate, in accordance with the formula (*). If we proceed thus through n stages, we obtain for n > 1 / \ ^ nx \ *. n x i 2 v-T 1 f M. n ( x + v ) i n ( x v )l ^ n & I (f) rco;cot,r3 = ^{cot + ^ [ COt ^2 + COt 2^ J "" tan 2^" Now by 115, lim ,? cot z = 1 2->0 and hence for each a =[= v 1 , a 1 if in the above expression we letw*oo and, a^ /*Vs^ tentatively, carry out the limiting process for each term separately, we obtain the ex- pansion We proceed to show that this in general faulty mode of passage to the limit has, however, led in this case to a right result. We first note that the series converges absolutely for every x -\~ 1 2, ..., by 70, 4, since the absolute values of its terms 206 Chapter VI. The expansions of the so-called elementary functfons. are asymptotically equal to those of the series JJ-^ . Now choose an arbitrary integer k > 6 | x \ , to be kept provisionally fixed. If n is so large that the number 2 n ~ 1 1, which we will denote for short by m, is > k, we then split up the expression (f) for nxcotx, as follows 35 : 71X ( ^- J (In the square brackets we have of course to insert the same expression as occurs in (f).) The two parts of this expression we denote by A n and B n . Since A n consists of a finite number of terms, the passage to the limit term by term is certainly allowed there, by 41, 9, and we have lim A n = 1 + 2 x* JS-r^-T-. ->. M r^i* J -* f Also B n is precisely nx coinx A n , hence lim B n certainly exists. Let r k denote its value, depending as it does upon the chosen value ft; thus limB r * i i n = r k = n x cot n x 1 + 2 x* ~i L v=l X ~~ V " J Bounds above for the numbers B n , for their limit r k and so finally for the difference on the right hand side, may now quite easily be esti- mated: We have * / i IA i M. / t\ 2 cot a cot (a + b) + cot (a-b) = - . - ff sin* a and hence --.^(z v) 2 cot a * writing for the moment ^ = a and ^ = B 9 for short. As 2 n > ft > 6 | a |, we certainly have | a | = itx < 1 and so 26 I sm a I = Since, further, < ft < "~ < 2, we have 27 86 Cf. Footnote 7, p. 194. 86 For the sake of later applications we make these estimates in the above rough form. 87 Cf. p. 200. 24. The trigonometrical functions. 207 Hence sin ft sin a 61*1 the latter, because r > ft > 6 cc | . It therefore follows that (for v > k) cot- and hence 36 x- 72 ** -1 The factor outside the sign of summation is quite roughly estimated certainly < 3; for 1 2 cot 2 1 = Accordingly, nx was < 1, and for | z \ < 1 we have _ 31 5! 1 But this is a number quite independent of w, so that we may also write But the bound above which we have thus obtained for r k is equal to the remainder, after the k ih term, of a convergent series 28 . Hence r k -~> as k -> + oo. If we refer back to the meaning of r k , we see that this implies lim { 77 x cot 77 x - [l + 2 x 2 2 ^ l _ J } r- 0, or, as asserted, a formula which is thus proved valid for every x ^ 0, + 1, 2, . . . . 6. We shall in the next chapter but one make important applications (p. 236 seqq.) of this most remarkable expansion in partial fractions, as it is called, of the function cot. We can of course easily deduce many further such expressions from it; we make note of the following: 1 The convergence is obtained just as simply as, previously, that of the series V - 208 Chapter VI. The expansions of the so-called elementary functions. The formula rccot^ 2^cotjra; = jrtan~ first of all gives 29 fft tan >t ass, % # ^ :t 1 ~t~ 3 , + 5 . < 2 .~ r9 v _L n 2 __/** -r -i- j- > j. * = 2 The formula >V_J _J \ rlToM 2 "* 1 )-* (2 v + l)+*/' 2 sin * then gives further, for a;=j=0, 1, 2, ... 7T 1 9 'Y* 9 'V* ^ = -- I * x *' r I . .. sin jt x x ' i ^ ^ o^ * Finally if we here replace x by \ re, we deduce 3-2*7 \3 + 2a; 5 - 2 W ' By 83, 2, Supplementary theorem, the brackets may heie be omitted. But if we then take the terms together again in pairs, starting from the beginning, we obtain, provided x ={= |, o> S> COSTTA: With these expansions in partial fractions for the functions cot, tan, 4-andJ sm c< functions. -r- and , we will terminate our discussion of the trigonometrical sm cos 25. The binomial series. We have already, in 22, seen that the binomial theorem for positive integral exponents, if written in the form remains unaltered in the case 30 of a negative integral k. But we have then to stipulate | x | < 1. We will now show that with this restriction 99 The formula first follows only for x =)= 0, 1, 2, ... but can then be verified without any difficulty for o? = 0, 2, 4, .... (The series has the sum 0, as is most easily seen from the second expression, for an even integral x.) 80 In the former case the series is infinite only in form, in the latter it is actually so. 25. The binomial series. 209 the theorem holds 31 even for any real exponent a, i. e. , , (a any real number. As in the preceding cases, we will start from the series and shew that it represents the function in question The convergence of the series for [ x \ < 1 may be at once established; for the absolute value of the ratio of the (n + l) th to the n th term is and therefore >|a;|, which by 76, 2 proves that the exact radius of convergence of the binomial series is 1. It is not quite so easy to see that its sum is equal to the of course positive value of (1 -|- #) a If we denote provisionally by f a (x) the function represented by the series for | x \ < 1, the proof may be carried out as follows. Since 57 ( a \x n converges absolutely for I x I < 1, whatever may be \n / ' l the value of a, it follows, by 91, Rem. 1, that for any a and ^, and every \x\ < 1, we have Now as may quite easily be verified e. g. by induction 33 . Hence for 81 The symbol f a J is defined for an arbitrary real a and integral n j> by the two conventions (:)- and for every real a and every n > 1 , it satisfies the relation, which may at once be verified by calculation. 14 For this, give the statement, by multiplying by n\, the form F Z ~l7^(-^ Then multiply each of the (n -f 1) terms on the left hand side first by the corres- ponding: term of or, (a- 1), . .., (a-), ..., (a-w), then by the corresponding term of 0?-n), (/J-M + 1), ..., (fi-n + k) ..... f and add, so that in all we multiply by (-f /? n)\ grouping together the similar terms on the left, we obtain precisely the asserted equality, where n is replaced by w-j-1. The above formula is usually called the addition theorem for the binomial coefficients. 210 Chapter VI. The expansions of the so-called elementary functions. fixed | x | < 1, we have, for any a and f! 9 fa'fp = A+0- By precisely the same method as we used to deduce from the theorem of the exponential function, E (xj E (x t2 } E(x that for every real x we had (E (x) = (E (I)/, so we could conclude that for every a, /- = (/;)", if we knew here also that f a was for every real cc (with fixed x) a continuous function of a. As f 1 = l-{-x, the equality fa - (1 + *) would then be established generally for the stated values of x. The proof of the continuity results quite simply from the main rearrangement theorem 9O : If we write the series for f a in the more explicit form (a) /L _ 1 + , + (^_)^ + (^_^ + |.) a - + ... and then replace each term by its absolute value, we obtain the series also convergent for | x \ <C 1 by the ratio test. We may accordingly rearrange the above series (a) in powers of cc, obtaining (b) fa =i i. e. certainly a power series in cc. Since this still for fixed x in | x | < 1 converges, by the manner in which it was obtained, for every cc> we have an everywhere convergent power series in a, hence certainly a continuous function of cc. This completes the proof 33 of the validity of the expansion 119 and at the same time fills the gap left in the proof of the reversion theorem 21. 83 An alternative proof, perhaps still easier than the above, but using- the 00 /#\ differential calculus, is as follows: From f a (x) = J { ) x n it follows that n=0 vw/ Since however (" + 1 ) ( + j) = (" ) ' U follows further that KM = /'-! <*) But (1+ *) /a-! (*) = (1 + *) ' 26. The logarithmic series. 211 The binomial series provides, like the exponential series, an expansion of the general power a?: Choose a (positive) number c for which, on the one hand, tf may be regarded as known, and on the other, < - < 2. a Then we may write c = 1 + x with | x \ < 1 and so obtain, as the required expansion, * = *(!+*)* = T* l + X Thus e. g. = 5 L 1 ~~ \ 1 / 60 "f~ \ 2 ) 50-' ~~~ \ 3 / 50* ^ ' * * J is a convenient expansion of V2. The discovery of the binomial series by Newton M forms one of the landmarks in the development of mathematical science. Later Abel 35 made this series the subject of researches which represent a perhaps equally important landmark in the development of the theory of scries (cf. below 170, 1 and 247). 26. The logarithmic series. As already observed on pp. 58 and 83, in theoretical investigations it is convenient to employ exclusively the so-called natural logarithms, that is to say, those with the base e. In the sequel, log* shall therefore always stand for log e x (x > 0). If y = log x, then x = e v or By the theorem for the reversion of power series (107), y = log x is there- thus, for every | x \ < 1, we have the equation (1 + *)/'<*)-/.<*) = <>. Since (1 + x)* > 0, this shows that the quotient has everywhere the differential coefficient 0, i. e. is identically equal to one and the same constant. For x ~ the value is at once calculated and = -|- 1 ; thus the assertion /a (x) = (1 + jc) a is proved afresh. 34 Letter to Oldenburg, 13 June 1676. Newton at that time possessed no proof of the formula; the first proof was found in 1774 by Euler. 86 J. f. d. reine u. angew. Math., Vol. 1, p. 311, 1826. 212 Chapter VI. The expansions of the so-called elementary functions. fore expansible in powers of (x - 1) for all values of x sufficiently near to + 1, or y = log (1 + x) in powers of x, for every sufficiently small | x \ : y - log (1 + *) = * + ft, a* + ft 3 x* + . . . . The coefficients b n may actually be evaluated by the process indicated, provided the working is skilfully set out 36 . But it is advisable to seek more convenient methods: For this purpose, the developments of the preceding section suffice. For | x \ < I and arbitrary a, the function / a = / a (x) there examined is Using, for the left hand side, the expression (b) of the former paragraph and for the right hand side, the exponential series, we obtain the two power series everywhere convergent: = 1 + [log (! + *)]<*+... . By the identity theorem for power series 97, the coefficients of corre- sponding powers of a must here coincide. Thus, in particular 37 , and for every | x \ < 1 120. (a) log (1 + x) - x - f + f - + . . . 4 ( -^P *+... Thus we have obtained the desired expansion, which, we also see a pos- teriori^ cannot hold for | x \ > 1 . If we replace in this logarithmic series, as it is called, x by x and change the signs on both sides of the equality, we obtain, equally for every | x \ < 1, By addition we deduce, again for every | x \ < 1 , There are of course various other ways of obtaining these expansions; but they either do not follow so immediately from the definition of the log as inverse function of the exponential function, or make more extensive use of the differential and integral calculus 38 . 36 Herm. Schmidt, Jahresber. d. Deutsch. Math. Ver., Vol. 48, p. 56. 1938. 87 Cf. the historical remarks in 69, 8. 88 We may indicate the following two ways: 1. We know from the reversion theorem that we may write log (1 H- x) = x + 6 2 * a + 6 3 x 3 -f . . . ; it follows from Taylor's series 99 that = 1 (& log (1 + x)\ ^(-l)*- 1 -Q k 27. The cyclometrical functions. 213 Our mode of obtaining the logarithmic series also the two modes mentioned in the footnote do not enable us to determine whether the representation remains valid for x = -f- 1 or x = 1. Since however 120a reduces, for x = + 1> to tne convergent series (v. 81 c, 3) the value of this series, by Abel's theorem of limits, is = lim Iog(l + #) = log2. jc-yl .0 Our representation (a) there remains valid for x = -f- 1; but for x -= it certainly no longer holds, as the series is then divergent. 27. The cyclometrical functions. Since the trigonometrical functions sin and tan are expansible in power series in which the first power of the variable has the coeffi- cient 1, different from 0, this is also true of their inverses, the so- called cyclometrical functions sin" 1 and tan" 1 . We have therefore to write, for every sufficiently small | x \ , y = sin" * x == x -f- & 3 or* + & 5 # 5 + * y = tan" * x = x + & 3 ' x* + 6 5 f a; 5 -| ---- where we have left out the even powers at once, since our functions are odd. Here too it would be tedious to seek to evaluate the coefficients b and b f by the general process ot 107. We again choose more con- venient methods: The series for tan"" 1 a; is the inverse of v _?? 4.^-4.... , , . siny y 31"*" 5! "*" (a) x = tan y = -- = -- ;, - ; -- , ^ / ^ cos 1 * ' cosy or of the series obtained by 1O5, 4 after carrying out the process of division in the last quotient. If here all the signs, in numerator and denominator, were -J-, then we should be concerned with reversing the function 2 . = T =l-* + *^ It Integrating, it follows at once, by O r theorem 5, since log 1=0, that The method in the text is so far simpler that it proceeds entirely without the use of the differential and integral calculus. 8 (051) 214 Chapter VI. The expansions of the so-called elementary functions. But the inverse of this function is, as we immediately find, By the general remark at the end of 21, the reverse series of the series for x = tanjy actually before us is obtained from the series last written down by alternating the signs 39 again, i. e. 121. tm~ l x = *- + ?-+ . If therefore this power series, which obviously has the radius of conver- gence 1, is substituted for y in the quotient on the right of (a), and this is then rearranged, as is certainly allowed, we obtain the terminating power series x. Hence its sum for an arbitrary given \ x \ < 1 is a solution of tan y = x, and is precisely the so-called principal value of the function tan" 1 x hereby defined, i. e. the value which is = for x = and then varies continuously with x. Hence for !<#< + !, it satisfies the condition and is defined, in the interior of this interval, without any ambiguity. For \x\ >1 the expansion obtained is certainly no longer valid; but Abel's theorem of limits shews that it does still hold for x 1. For the series remains convergent at both endpoints of the interval of convergence and tan" 1 x is continuous at both these points. We have therefore in particular the series, peculiarly remarkable for clearness and simplicity: _ _ _ 4 A 3^5 7 T- giving at the same time a first means of determining TT of some practical value. This beautiful equation is usually named after Leibniz w \ it may be said to reduce the treatment of the number IT to pure arithmetic. It is as if, by this expansion, the veil which hung over that strange number had been drawn aside. 89 A different method is the following: We have d tan" 1 x 1 _ 1 __ 1 _, 2 r 4 i ~d*~- ?^~r+i^;~i-f * 2 * +* * dy the latter for | x \ < 1. As tan" 1 = 0, it follows by 99, theorem 5, that for | x \ < 1, x* tan' 1 *-*- 3 +-5 -+ A method corresponding to that given first in the preceding footnote is some- what more troublesome here, as the differential coefficients of higher order of tan" 1 x even at the single point are not easy to find directly. The expansion of tan" 1 x was found in 1671 by J. Gregory, but did not become known till 1712. 40 He probably discovered it in 1673 from geometrical considerations and without reference to the inverse tan-series. Exercises on Chapter VI. 215 For the deduction of a series for sin"" 1 a;, the method which we have just used for tan"" 1 a; is not available. The process indicated in the last footnote, however, provides the desired series: We have for x < 1 _ dx /d*>iny\ cosy i/l x 9 (~dJ the positive sign being given to the radical since the derived function of sin" 1 a; is constantly positive in the interval 1 . . . -f- 1. From (sin-' *)' = 1 - (-') ** + (-) * - + it at once follows, however, by 99, theorem 5, as sin~ A = 0, that for x < 1 sin l x 1 cc 3 . 1-3 a? 5 , 1-3-5 2 T + 53TT + 2T* This power series also has radius 1, and on quite similar grounds to the above we conclude that for | x | < 1 its sum is the principal value of sin" 1 a;, i. e. that uniquely determined solution y of the equation sin y = x which lies between -^ and + TT For #= 1, the equality is not yet secured. By Abel's theorem of limits it will hold there if, and only if, the series converges there. As we have a mere change of sign in passing from -}- x to x, this only needs testing for the point -f- 1 . There we have a series of po- sitive tcims and it suffices to show that its partial sums are bounded. Now for < x < 1, if we denote by s n (x) the partial sums of 123, s n (x) < sin" 1 a; < sin~ 1 l = -|. And as this holds (with fixed n) for every positive x < 1, we also have and as this holds for every n, we have proved what we required. Thus i-l + i.i + ld.i + LM.i-J-... 2 ~2 3~2-4 5^2-4-6 7 ' ' 22 to 27 have thus put us in possession of all the power series which are most important for applications. Exercises on Chapter VI. 74. Show that the expansions in power scries of the following functions have the form indicated in each case: a) tf^sina^ x n with s = V^" sin n T * e - 5 U = , n=o n! ,^(-l)*- l b k -~ with b k =l + ~ + l 216 Chapter VI. The expansions of the so-called elementary functions. c) -tan-^-lo 115= yT c .,g 4 * +2 w ;.u ,._i j . * . . * d) ~tan-i o:. with ^ = 1 + 3 +. <j i r il 3 /* _ ) log- - = JS, 7 n ~ 1 a; w , with the same meaning- of h n as in d). 2 L 1 an n==2 n 75. Show that the expansions in power series of the following functions begin with the terms indicated: x __ 1 x y? a: 3 Io *ri; x +*+. ..+!! b) (I-*)/ 2 m = 1 1 29 c) tan (sms)-sm(tana;) = gga; 7 + ^a; H ; ~ , a; .11 7 , . 2447 A 959 76. Deduce, with reference to 1O5, 5, 115 and 116, the expansions in power series of the following functions a) log cos x ; b) log ; x x , tana; _. x c) log ; d) -; ; ' 1 cos x f cos x ' X 1 2 sin x & ' e^+1 ' cosx sin* 77. Show that, for a =t= 0, 2, 4, ... 78. We have (^ -- -^} *>e. Is the sequence monotone? Increasing or decreasing? What, in this respect, is the behaviour of the sequences (n-i)-*', o< a <i ? 70. From x n -* f it invariably follows that Exercises on Chapter VI. 217 and also, if x n and are positive, that x * 8O. If (#) is an arbitrary real sequence, for which ->0, and we write n Cx \ n 1 -- -J =y n , then, in every case, 81. Prove the inequalities of 114. 8S. Express the sums of the following series by closed expressions in terms of the elementary functions: , 1 x x* x* aj 2" f "5 + T + ll + ""* (Hint: If f(x) be the required function, then obviously (**./>)/ = 1-^ whence f(x) may be determined. Similarly in the following 1 examples.) ~3 ~5 r ? ~9 M x _ x 4. _ __ _ 4. _ . . . ' 1-3 3.5^5.7 7-9" 1 " 1 X X* C ) 1.2.3 + 2-3 4 + S'^S 4 " "" ; 83. Obtain the sums of the following series as particular values of ele- mentary functions: . . _ "2 2^4 2 4-6 2 4-6-8 " ' " ~ ' I _u 1<3 . l _iL 5 i 7 __ . ^Ji 3 : 5 :. 7 :^- n __ 4. _ ! AO . C) 2 + 2 4 + 2 4-6-8 10 + 2 4 6-8.10.12.14 + 2 v ' dN - 1 1>8 1-3 5-7 1 3-5-7 9 H ^ 2 2 4 6 + 2- 4-6 8 10 2 4-6 8~l6-l2""u 84. Deduce from the expansion in partial fractions 117 seq. the following expressions for n: -- l - l - [ - -I f - ~ Vl - 2T-! + 2 aV"T + ~ ~ + + where a=t=0, l,i, ij,... Substitute in particular a - 3, 4, 6. 218 Chapter VII. Infinite products. Chapter VII. Infinite products. 28. Products with positive terms. An infinite product is, by 11, II, to be taken merely as representing a new symbol for the sequence of the partial products */! 2 . . . u n . Accordingly such an infinite product should be called convergent, with value U y oo if the sequence of the partial products tends to the number U as limit. But this is particularly inconvenient, owing to the fact that then every product would have to be called convergent for which a single factor was = 0. For if u m were 0, then the sequence of partial products also would tend to U = 0, since its terms would all be equal to for n ^ m. Simi- larly every product would be convergent again with the value for which from some m onwards In order to exclude these trivial cases, we do not describe the behaviour of an infinite product by that of the sequence of its partial products, but adopt the following more suitable definition, which takes into account the peculiar part played by the number in multiplication: 125. o Definition. The infinite product CO L * n = U^ Uq ' Uft . n=i will be called convergent (in the stricter sense) if from some point onwards say for every n > m ~ no factor vanishes, and if the partial products, beginning immediately beyond this point tend, as n increases, to a limit, finite and different from 0. // this be = f/ w , then the number U = u l .u.>-....u m -V m , obviously independent of m, is regarded as the value of the product*. 1 Infinite products are first found in F. Vieta (Opera, Leyden 1646, p. 400) who gives the product 28. Products with positive terms. 219 We then have first, as for finite products, the o Theorem 1. A convergent infinite -product has the value if, and only if, one of its factors is = . As further p n -i~-+U m with p n +U m> and as U m is 4= 0, we have (by 41,11) u = -*i *1 W Pn-l and we have the o Theorem 2. The sequence of the factors in a convergent infinite product always tends >1. On this account, it will be more convenient to denote the factors by u n = 1 -f- a n , so that the products considered have the form // (! + ) n=i For these, the condition a n > is then a necessary condition for con- vergence. The numbers a n as the most essential parts of the factors will be called the terms of the product. If they are all ^> 0, then as in the case of infinite series, we speak of products with positive terms. We will first concern ourselves with these. The question of convergence is entirely answered here by the Theorem 3. A product 77 (1 -f- a n ) with positive terms a n is convergent if, and only if, the series a n converges. Proof. The partial products p n = (1 -J- t )- (1 + # n )> since a n ^ increase monotonely; hence the First main criterion (46) is available and we only have to show that the partial products p n are bounded if, and only if, the partial sums s n = a t + > ~l ---- + # are bounded. Now by 114 a, 1 -f- a v <[ e a >" and so for each n p n ^e'> on the other hand the latter because in the product, after expansion, we have, besides the terms of s n , many others, but all non-negative ones, occurring. Thus for each n (cf Ex.89) and in /. Walhs (Opera I, Oxford 1695, p. 468) who in 1656 gives the product - ~2~~~T'T' ~3~ 5 5 7 But infinite products first secured a footing- in mathematics through Enter, who established a number of important expansions in infinite product form. The first criteria of convergence are due to Cauchy. 220 Chapter VII. Infinite products. The former inequality shows that p n remains bounded when s n does, the latter, conversely, that s n remains bounded when p n does, which proves the statement. 9 Examples. 1. As we are already acquainted with a number of examples of con- vergent series 2 a n with positive terms, we may obtain, by theorem 3, as many examples of convergent products J7(l+0 n ). We may mention: (l +~) is convergent for a. > 1, divergent for a <J 1. The latter is more easily recognised here than in the corresponding series 3 , for 77" (l 2. H(l+x n ) is convergent for 0<#<1; similarly 0* 00 Q. m i I __ . I g=s I i rs - . ^ \ W (W -{~ I)/ -*--M. fi foi ~| j) Q With theorem 3 we may at once couple the following very similar Theorem 4. //, for every n, a n ^ 0, then the product //(I fl w ) also is convergent if, and only if, 2 a n Converges. Proof. If a n does not tend to 0, both the series and the pro- duct certainly diverge. But if a n * 0, then from some point onwards, say for every n > m, we have a n < \, or 1 a n > ^. We consider the series and product from this point onwards only. Now if the product converges, then the monotone decreasing sequence of its partial products p n = (1 a <m + i)*"(l O tends to a positive (> 0) number U m , and, for every n > m, Since, for <C a r <C 1, we always have (as is at once seen by multiplying up), we certainly have \ ' in T* I/ V i WIT 2' V n/ ^J 9 In the first part of the proof of this elementary theorem, we use the transcendental exponential function. We can avoid this as follows: If converges, choose m so that for every n > m , , , . 1 <* m + i -f a m + 8 H h a n < --- . As, obviously, for these w's, we now have (1 -f a +1 )--- (l-f-a)<l-l-K- M + - -f K + i + - we certainly have, for all w's, hence (/>) is bounded. 8 In this we have therefore, on account of theorem 3, a new proof of the divergence of 5] . 29. Products with arbitrary terms. Absolute convergence. 221 Accordingly the convergence of the product 77(1 -f- a n ], and hence of the series 2 a n , results from that of 77(1 a n ). If, conversely, 2 a n converges, then so does S 2 a n , and consequently by Theorem 3 the product 77 (1 + 2 a n ) also does. Hence, with a suitable choice of K, the products (1 + 2 a m+1 ) . . . (1 + 2 a n ) remain < K. If we now use the fact that, for ^ a v ^ , 1 - a v ^ r+ 2 as may again be seen by multiplying up we infer and the partial products on the left hand side, as they form a mono- tone decreasing sequence, therefore tend to a positive limit: i. e. the product 77(1 <z n ) is convergent. Remarks and Examples. 126 00 ! // I 1 ) is convergent for a > 1 , divergent for a< 1. .-=. l "' 2. If a n < 1 and if 27 a n diverges, then II (I a n ) is not convergent, with our definition. As however the partial products p n decrease monotonely and remain > 0, they have a limit, but one which is necessarily = 0. We say that the product diverges to 0. The exceptional part played by the number thus involves us in some slight incongruity of expression. A product is called divergent whose partial products form a decidedly convergent sequence, namely a null sequence, (p n ). The addition "in the stricter sense" to the word "convergent" in Def. 125 is intended to serve as a reminder of this fact. 3. That e. g. JJ M -- J diverges to is again very easily seen from n=2 29. Products with arbitrary terms, 4 Absolute convergence. Jf the terms a n of a product have arbitrary signs, then the following theorem corresponding to the second principal criterion 81 for series holds: o Theorem 5. The infinite product 77(l + a n ) converges if, and 4 A lull and systematic account of the theory of convergence of infinite products may be found in A. Pringsheim: t)ber die Konvergenz unendlicher Produkte, Math. Annalen, Vol.33, p. 119 154, 1889. 222 Chapter VII. Infinite products. only if, given e > 0, we can determine 5 n Q so that for every n > n and every k ^> 1, [(1 + . + 1 )(l + .+.) (! + . + *) ~ 1] < * Proof, a) If the product converges, then from some point on- wards, say for every n > m, we have a n ={= 1, and the partial products tend to a limit 4= 0. Hence there exists (v. 41, 3) a positive number /f such that, for every n > m, \p n \ ^ /? > 0. By the second principal criterion 49 we may now, given e > 0, determine w so that for every n > n and every k ^> 1, ln +fc -M< e '/*- But then, for the same n and &, which is precisely what we asserted. b) Conversely, if the e- condition of the theorem is fulfilled, first choose e = |, and determine m so that, for every n > m t For these n's we then have showing that, for every n > m, we must have l-f-a w =j=0; and further, that if p n tends to a limit at all, this certainly cannot be 0. But we may now, given e > 0, choose the number n so that for everv n > n Q and every k ^> 1, ": _ 1 Pn or And this shows that p n really has a unique limit. Thus the conver- gence of the product is established. As in the case of infinite series, so similarly in that of infinite products, those are the most easily dealt with which converge "abso- lutely". By this we do not mean products IT u n for which H\u n \ also converges, such a definition would be valueless, since then every convergent product would also be absolutely convergent, but we define, on the contrary, as follows: 127. o Definition. The product 77(1 + a n ) is said to be absolutely con- vergent if the product 77(l + |a n |) converges. 6 Or v. 81, 2nd form if invariably f (1 + a n + 1 ) (1 -f * + *) . (1 + a n 4.^)] _ ! . or v. 81, 3 rd form if invariably [(1 + +!) ..(H- ,+*)] -*1. 29. Products with arbitrary terms. Absolute convergence. 223 This definition only gains significance through the theorem: o Theorem 6. The convergence of 17(1 -f- | a n | ) involves that of + O- Proof. We have invariably I (1 + + ,)(! + +,) .(1 + +)-!! as is at once verified by multiplying out. If therefore the necessary and sufficient condition for the convergence of Theorem 5 is satisfied by 77(1 -|- | a n \), it is ipso facto satisfied by 77(1 + a n ), q. e. d. In consequence of Theorem 3, we may therefore at once state o Theorem 7. A product 77(1 + a n ) is absolutely convergent if, and only if, a n converges absolutely. As we have an already sufficiently developed theory for the determination of the absolute convergence of a series, Theorem 7 solves the problem of convergence in a satisfactory manner for absolutely convergent products. In all other cases, the following theorem reduces the problem cf convergence of products completely to the correspond- ing one for series: Theorem 8. The product 77(1 -f- O converges if, and only if the series ^ 2 log (! + ) n=w-i-l commencing with a suitable index*, converges. And the convergence of the product is absolute if, and only if, that of the series is so. Furthermore, if L is the sum of the series, then n=L Proof, a) If 77(1 -f- a n ) converges, then a n >0 and hence from some point onwards, say for every n > m, we have | a n \ < 1. Since, further, the partial products = (!+ , + i^ (! + )> (n > }> tend to a limit U m 4= (hence positive), we have by (42, 2), But log/> n is the partial sum, ending with the th term, of the series in question. This, therefore, converges to the sum L = log U m . As U e L , we thus have b) If, conversely, the series is known to converge, and to have the sum L, then we have precisely log p n >L, and consequently (by 42, 1) This completes the proof of the first part of the theorem, since <? e It suffices to choose m so that for every n ;> m we have | a m \ < 1 . 224 Chapter VII. Infinite products. To deduce, finally, that the series and product are, in every possible case, either both or neither absolutely convergent, we use with theorem 7 and 70, 4, the fact that (112, b), when a n ->0 >* a + ) I , (Here any terms a n which = may be simply omitted from con- sideration.) Although we have thus completely reduced the problem of the convergence of infinite products to that of infinite series, yet the result cannot entirely satisfy us, because of the difficulties usually involved in the practical determination of the convergence of a series of the form J^log (1 + a n ). The want here felt may, at least partially, be supplied by the following Theorem 9. The series (starting with a suitable initial index) JHog (1 -j- a n ) and with it the product 77(1 -j- a n ), is certainly convergent, if 2 a n converges and if 2a^ is absolutely convergent 7 . Proof. We choose m so that for every n ~> m, we have \a n \ < ~, and consider 77(1 + a n ) and 2 log (1 -|- aj, starting with the (m + l)* h terms. If we write then the numbers # M so determined certainly form a bounded sequence, for 8 , as a n ->0, ft n ~ > J. If therefore 2 a n and J?|0 n | a are con- vergent, J?log(l + fl n ), and hence also 77(l-|-a n ), is convergent. This simple theorem leads easily to the following further theorem Theorem 10. If2a^ is absolutely convergent, and \a n \ is < 1 for eveiy n > m, then the partial products n n p n = J[ (1 + 0J and the partial sums s n = a v (n > m) } v=wfl v=m-H are so related that p n s^, e*n i. e. the ratio of the two sides of this relation tends to a definite limit, finite and =f= 0> whether or no 2a n converges. 7 J*fl rt 2 , if convergent at all, is certainly absolutely convergent. We adopt the above wording so that the theorem may remain true for complex a n 's, for which n 2 is not necessarily ;> (cf. 57). 8 For < | x | < 1 we have in fact or log(l + ) - * _ _ 1 x i> " " Y + 3"~ h "* And those terms which are possibly = may be again simply neglected, as they have no influence on the question under consideration. 29. Products with arbitrary terms. Absolute convergence. 225 Proof. If we adopt the notation of the preceding proof, then, as log (1 + a n ] = a n -|- $'&, we have for every n > m (l + m+1 ) -(! + )= A vm+l if the sums in the last two exponents are taken also from v = m -f- 1 to v = n . And as 2$ n a^, the $ n 's being bounded, converges absolutely when -0 n a does so, we can, from the above equation, at once infer the result stated. This theorem also provides the following, often useful Supplementary theorem. // 2a n * converges absolutely, then 2a n and /7(1 -f- a n ) converge and diverge together. Remarks and examples. 128. 1. The conditions of Theorem 9 are only sufficient', the product 77(1 + n ) may converge, without 2 a n converging. But in that case, by Theorem 10, v I a la must also diverge. / 1 \ 2. If we apply theorem 10 to the (divergent) product 7/1 H -- ) , then it n=i v n ' follows that e h n ^ n if h n denotes the w 01 partial sum of the harmonic series A n = l-| --- h H -- - 2 n e h n Accordingly the limits lim - = c and lim [h n log n] = log c = C exist, the latter because c ^= 0, hence > 0. The number C defined by the second limit is called Ruler's or Mascherom's constant. Its numerical value is C = 0-577 215 6649 . . . (cf. Ex. 86 a, 176, 1 and 64, B, 4). The latter result gives us further valuable information as to the degree of divergence of the harmonic series, as it gives h n ^ log n . Further the estimates of bounds above made for the proot of Theorem 3 show, even more precisely, if we there put a y = , that or h n > A J| _ 1 > log n so that Euler's constant cannot be negative. QO / / n-i \ 3- 7/ U + - " - ) * s convergent. Its value may, as it happens, be n=i X n ' found at once by forming the partial products, and is = 1 . that 00 / x \ 4. H[l-i -- J diverges for x ={= 0. However, theorem 10 shows v n/ M n ' , or what is the same thing by 2, ^^ n*, tO/ y,\ ]l ( 1 H -- J ^ v * i. e. (v. 40, def. 5) the ratio 226 Chapter VII. Infinite products. has, for every (fixed) a; , when M-KX>, a determinate (finite) limit which is also different from if a; is taken =j= 1, 2, ... (cf. below, 219, 4). oo / 3.3 \ 5. II I 1 -1 is absolutely convergent for every x. n=l V w ' 30. Connection between series and products. Conditional and unconditional convergence. We have more than once observed that an infinite series 2 a n is merely another symbol for the sequence (sj of its partial sums. Apart from the fact that we have to take into account the exceptional part played by the value in multiplication, the corresponding remark holds good for infinite products. It follows that, with this reservation, every series may be written as a product and every product as a series. As regards detail, this has to be done as follows : 129. 1. If ZT(l ~t" a n) * s &i ven > t ^ ien th* 3 P r duct, if we write H (! + ) = />> represents essentially the sequence ( n ). This sequence, on the other hand, is represented by the series This and the given product have the same meaning if the product converges in accordance with our definition. But the series may also have a meaning without this being the case for the product (e. g. if the factor (l -f- # 5 ) is = and all other factors are = 2). 00 2. If conversely the scries a n is given, then it represents the n=l sequence for which s n = J^<V This is also what is meant by the r = l product .*..*... . 7^_ln ^. . fiL , _ __ \ Sl * * 2 =Sl n iVW" * ni\ [ > + ^ + . + -"-M n - 1 '' provided it has a meaning at all. And for this obviously all that we require is that each s n 4= 0. In general the convergence of the product implies the convergence of the series, and conversely. In the case, however, of s n -> 0, although we call the series convergent with sum 0, we say that the product diverges to 0. ^ ^ Thus e. g. the symbols E ^ and ~ // (l + o^rio =-! * = X * JUVi) have precisely the same meaning. 30. Connection between series and products. 227 It is, however, only in rare cases that a passage such as this from the one symbol to the other will be advantageous for actual investigations. The connection between series and products which is theoretically conclusive was, moreover, established by Theorem 8 alone, or by Theorem 7, if we are concerned with the mere question of absolute convergence. In order to show the bearing of these theorems on general questions, we may prove as analogue of Theorem 88, 1, and 89, 2, the following: Theorem 11. An infinite product 77(1 + 0J is unconditionally 130 convergent i. e. remains convergent, with value unaltered, however its factors be rearranged (v. 27, 3) if, and only if, it converges absolutely 9 . Proof. We suppose given a convergent infinite product 77(1 + # n )- The terms a n , certainly finite in number, for which |0 n |^>, we re- place by 0. In so doing, we only make a "finite number of alterations" and we ensure | a n \ < \ for every n. The number m in the proof of theorem 8 may then be taken ~ 0. We first prove the theorem for the altered product. Now, with the present values of a u , /7(1 + J and 2-log(l + O are convergent together, and their values U and L stand in the relation U = e L to one another. It follows that a rearrangement of the factors of the product leaves this convergent, with the same value 17, if and only if the corresponding rearrangement of the terms of the series also leaves this convergent, with the same sum. But this, for a series, is the case if, and only if, it converges absolutely. By theorem 8 the same therefore holds for the product Now if, before the rearrangement, we have made a finite number of alterations, and then after the rearrangement make them again in the opposite sense, this can have no influence on the present question. The theorem is therefore true for all products Additional remark. Using the theorem of Riemann proved later (187) we can of course say, more precisely: If the product is not absolutely convergent and has no factor =0, then we can by suitable rearrangement of its factors, always arrange that the sequence of its partial products has prescribed lower and upper limits x and /*, provided they have the same sign as the value of the given product 10 . Here * and ft may also be or 00. 9 Dim, U.: Sui prodotti infiniti, Annali di Matem., (2) Vol. 2, pp. 2838. 1868. 10 For a convergent infinite product has certainly only a finite number of negative factors; and their number is not altered by the rearrangement. 228 Chapter VI I. Infinite products. Exercises on Chapter VII. 85. Prove that the following products converge and have the values indi- cated: ' / Zn + 1 \ 4 c) 77 f 1 + ( -^-nvqrijs ) = 3- n - 2 85a. By 128, 2 the sequences x n = I + J + + - T log n and y = 1 -fJ-h-.--f- 7i 1 n have positive terms for n > 1. Show that (# n | y n ) is a nest of intervals. The value so defined is Euler's constant. 86. Determine the behaviour of the following products: for a ^ 1, gt 3> 87. Show that IT cos ^ w converges if 2\ x n | a converges. 88. The product in Ex. 86 d has, for positive integral values of a, the value y^. [/ 1\ 2At^ jv_i Hint : The partial product with last factor ( 1 ^T ZT1 ) * s = H ( * --- ) \ / v-fc+iA ' - 89. Prove, with reference to Ex. 87, that cos v cos 5- cosy^ ... = -. 4 o lu 7T (We recognise Vieta's product mentioned in footnote 1, p. 218.) 90. Show, more generally, that for every x x x x x sin x t L i- t. cosh ^ cosh -. cosh r cosh T7i o lO JC g _j_ e a e _ e x in which latter formula cosh x = - , sinh ac = - denote the hyperbolic cosine and sine of x. 91. With the help of Ex. 90, show that the number defined by the nest of intervals in Ex. 8c is = "~^~^i where S- is defined as the acute angle for which cos $ = . Similarly the number defined by Ex. 8 d is =3 r x lf if & is defined by x i Exercises on Chapter VII. 229 92. In a similar way, show that the numbers defined in Ex. 8e and 8f have the values: e) . with O-T!*-* with 93. We have + - X( *~ - l) - + . . . + ( - I)- x (X What can you deduce for the series and product of which we here have the initial portions? 94. With the help of theorem 10 of 29, show that 1-3 5...(2n-Jl) J_ " 2.4.6...~2T~~~ ^ 95. Similarly, show that, for <j x < y , s(s+l)(s + 2)...(s + n) _^ y (y + 1) (y + 2) . . . (y + n) 96. Similarly, show that if a and b are positive, and A n and (7,, are respectively the arithmetic and geometric means, of the n quantities a, a-|-6, a + 2b, ..., a + (w-l)6, (n = 2, 3, 4, . . .) then A, * G n ~* 2 ' 97. What can be deduced, from the convergence of //(l-f-a w ) and &), as to lhat of (Cf. 83, 3 and 4.) 98. Given (u n ) monotone decreasing and * 1 , is 1 1 u -- 2 8 "4 M . --- u -- . 1 "2 8 "4 always convergent? (Cf. 82, theorem 5.) 99. To complete 29, theorem 9, prove that IJ(l-\-a n ) certainly con- verges if the two series S(a n -ta n *) and ^|a.| converge. How may this be generalized? On the other hand, show, by the example of the product where we assume J <; a < J , that 77(1+0*) may converge even when 2a n and ^a M 2 both diverge. 230 Chapter VIII. Closed and numerical expressions for the sums of series. Chapter VIII. Closed and numerical expressions for the sums of series. 31. Statement of the problem. In Chapters III and IV, we were concerned mainly with our problem A, the question of the convergence of series, and it was not till the last few chapters that we considered also the sum of the series. This latter point of view we shall now place in the fore- ground. It is necessary, however, in order to supplement our deve- lopments of pp. 78-79 and 105, that we should make it quite clear once more what is the significance of the questions which arise in this connection. If, for instance, we have proved the relation 122: i_i_ L , JL_JL+ 4 ~ 1 3^5 7 ^ we may interpret it in two ways. On the one hand, the equation indi- cates that the sum of the series on the right has the value ~ , one quarter of the value of a number 1 which we meet with in many other connections and to which approximations are well-known. In this sense, it may be claimed that we have specified the sum of the series written down above. But such a statement can only hold in a very relative sense; for it is not possible to give a complete specification of the number n, otherwise than by a nest of intervals or some equivalent symbol, and such a symbol is precisely furnished by the series; i. e. the expression on the right, in the above equation. We are therefore equally justified in claiming the exact opposite, namely that the equation provides an (extremely simple) expression for the number n in series form, that is to say, by means of a conver- gent sequence of numbers, which happens indeed in our case to have a peculiarly straightforward and convenient form and may also (69, 1) be immediately expressed as a nest of intervals 9 . The circumstances are entirely altered when we come to the equation (cf. 68, 2b): 1 1 In former times, when these matters were all interpreted rather geo- metrically, -^- was always thought of as the ratio of the area of a circle to that of the circumscribed square. * Namely: - = ( 5 2*| s a*+i)> where 31. Statement of the problem. 231 Here we are perfectly satisfied with the statement that the sum of the series is = 1, precisely because the number 1 (and similarly every rational number) can be fully and literally assigned. In such cases, we have a perfect right to assert that we have a closed expression for the sum of the series. But in all other cases, where the sum of the series is not a rational number, or at any rate not known to be one 3 , we cannot strictly speak of evaluating the sum of the series by means of a closed expression. On the contrary, the series ought then to be regarded as a (more or less imperfect) means of representing or approximating to its sum. By proceeding to express these approximations (usually in the form of decimal fractions) and estimating the errors involved, we form what is called a numerical evaluation of the sum. Lastly, as above in the case of the series for , we may have ascertained merely that the given series has for sum a number related in some simple (or at any rate specifiable) manner to a number which we meet with in other connections; as e. g. it follows from 122 and 124 that 1 1 - 3 In that case, we should still welcome the information so obtained, since it establishes a connection between results where formerly we saw none. It is usual, in such cases, still to say though in an extended sense that we have evaluated the sum by means of a closed expression; in fact, the number concerned is then regarded as "known" through those other connections, and we simply express the sum of the series "by means of a closed expression" involving this number. Here the student must, however, guard against self-delusion. If it has been ascertained, for instance (v. p. 211) that the sum of the series i+JL 1 4-L 3 . L + 1 ' 8 ".*.- 1 - L... 1 ' 2 50 ^~ 2-4 50* "^ 2-4-6 50 3 r has the value ^-Vs, it is still only in a very relative sense "deter- mined in the form of a closed expression". The number V2 is not per se any better known than the sum of any arbitrary convergent series. It is only because V2 occurs in so many hundreds of other connections and has, for practical purposes, been so often evaluated numerically, that we are in the habit of considering its value as almost as perfectly "known" as any literally specified rational number. If 3 For instance, if we have determined the sum of a series to be equal to Euler's constant, we do not know to this day whether we are confronted with a rational number or not. 232 Chapter VIII. Closed and numerical expressions for the sums of series. instead of the above scries, we consider, for instance, the following binomial series: _5_ [ 1 , J_ _24 4 24* _4 9 24 1 2 L 1+ 5 '1000 5-i6" 1000 a " 1 "5-10 15* 1000 s ' " J B, _ and its sum has been ascertained to be equal to y 100, we shall be less inclined to regard the sum as fully determined thereby; on the contrary, we shall prefer to accept the series as a most useful 5 means of evaluating y 100 to a degree of approximation not so easily attainable by other means. In other words, with the exception of those few cases in which the sum of a series can be specified as a definite rational number, when we consider equalities of the form "s = <<*", the emphasis will be laid sometimes on the right hand side and sometimes on the left, according to the circumstances of the case. If s may be considered as known through other connections, we shall still (though in an extended sense) say that the sum of the series has been evaluated in the foim of a closed expression. If this is not the case, we shall say that the series is a means of evaluating the number s (of which it provides the definition). (Obviously both points of view may be taken with regard to the same equality.) In the former of the two cases, we shall, so to speak, have achieved our object, since the problem B (v. p. 105) also is then solved to our satis- faction. In the latter case, however, a new task now begins, that of actually expressing the approximations, provided by the series itself, to its sum, in a convenient and simple form (e. g. in decimal fraction form, as the most desirable for our purposes), and of estimating the errors involved in these approximations. 32. Evaluation of the sum of a series by means of a closed expression. 1. Direct evaluation. It is obvious that we may without difficulty construct series with any assigned sum. If s be the assigned sum, construct, by any one of the many processes at our disposal, a sequence ( s n) conver gi n g to 5, and consider the series *o + (*i - *o) + (*, - O H h (* n s - 1) + ' ' ' Since its w th partial sum is precisely = s n , this series is convergent and has the sum s. This simple procedure affords an inexhaustible means of constructing series capable of summation in the form of a closed expression; e.g. we need only assume one of the numerous null sequences (a? n ) known to us, and write s n = s x n , n = 0, 1, 2, . ... Examples of series of sum 1. gives + 1+ < + --- = 1 32. Evaluation of the sum of a series by means of a closed expression. 233 ___ 1-2 2^ 34 4-5 6t V I * \ 1" * . w 7. If we multiply the terms of one of these series by 5, we obtain a convergent series of sum s. It is not superfluous to be able to construct such examples, as we shall see that the power to provide series with known sum is an advantage in the discussion of further series. The converse of the principle just treated is expressed by the Theorem. Given a series Jj a n , whose terms a n are expressible 131, in the form a n = x n x n + 1 , where x n is the term of a convergent sequence of known limit , the sum of the series can be specified, for we have ^> n = *o-- n=0 Proof. We may write Since x n *, the statement follows. Examples. 1. If a be any real number =f=0, l f 2,..., then (v. 68, 2b): 1 1 T 1 1 -^ == , as here 2. Similarly 2(a-f 1) as here i r i i 3. Generally, if p denotes any positive integer, 00 1 11 V = -- - . n^O ( K + M ) ( a + n + J ) ( + n "H P) P (a -f 1) . . . (a + p - 1) 4. Putting a = ^, we thus obtain, for instance, from 2.: rTr7 + 4"TTO + 7Tl OT 3 + * " = 24 ' 5. Putting a = 1 in 3. we obtain 1-2. .. or S, 1 = p + l n t?o(P + + l\ P \ P+l ) 234 Chapter VIII. Closed and numerical expressions for the sum& of series. The following is a somewhat more general theorem. 133. Theorem. // the term a n of a given series 2a n is expressible in the form x n x n + q , where x n is the term of a convergent sequence of known limit f , and q denotes a fixed integer > 0, then 2 a n = *o + *i H ----- h - 1 - ? * n=0 Proof. We have, for n > q> Since x v * ^ (by 41, 9), the statement at once follows. Examples. since here we have " q \a+n a- In particular, writing a = $ , oo 1 2 For a = l and q 2 we have accordingly: J_ J_ 1 ^^3. and for a = j, q = 3: _4._ + _4. ..- l.T^S-Q^.ll" 1 " "90" 3. Somewhat more generally, if A, as well as , denotes a fixed integer >0: 4. Thus for a = } , ^ = 2, A = 2 we find ^a. ^a. ^^ =i- 1- 5- 9 "*" 3- 7- 11 ^5- 9- 13"*"" 420* The artifices here employed may be extended to obtain, finally, the following considerably further reaching 134. Theorem. // the terms of a series 2 a n are expressible, for every n, in the form an = Ci x n + i + c t x n+* + '~ + c k x n+ic ( k constant, ^2) where (x n ) denotes a convergent sequence of known limit , and the coefficients C). satisfy the condition c i + c a H ----- f- c k =* 32. Evaluation of the sum of a series by means of a closed expression. 235 then 2 a n is convergent and lias for sum- JJ a n = c^x^ + (q + <g* a H ----- h fo + c,H ----- h * fc -iK-i n=o The proof is at once obtained by writing the expressions for fl 1? # a , ..., a m , one below the other so that terms involving x v occupy the same vertical. Carrying out the addition in columns, which of course is allowed even without reference to the main rearrangement theorem we find, for m > k, taking into account the condition ful- filled by the coefficients C A , which is again the sum of a finite number of terms. Letting m + oo, we at once obtain the required relation. Examples. n 2 1 Putting x n g - =- , & = 2, Cj = 1 , c s = -fl, we obtain J_ _5_ 7 2 M __ _^ '" 9 1 ^ > " 2" 1 / 1 1\ 13 27 - These examples may of course easily be multiplied to any extent desired. 2 Application to the elementary functions. The above few theo- rems have, speaking generally, made us familiar with all types of series which may, without requiring any more refined artifices, be summed in the form of a closed expression. By far the most frequent series, in all applications, are those ob- tained by substituting particular values for x in series expansions of elementary functions and in series derived from these by every species of transformation or combination, or other known processes of deduction. Examples, obtained in this manner, of summation by closed expres- sions are innumerable. We must content ourselves with referring the reader to the particularly ample selection of examples at the end of this chapter, in the working out of which the student will rapidly be- come familiar with all the main artifices used in this connection. The developments in this and the following section will afford further guidance in this part of the subject Let us merely observe quite generally, for the moment, that it is often possible to deal with a given series by splitting it up into two or more parts, each of which again represents a convergent series; or else by adding to or subtracting from 2 a n > term by term, a second scries of known sum. In particular, if a n is a rational function of n its expansion in partial fractions will frequently be a considerable help. 236 Chapter VIII. Closed and numerical expressions for the sums of series. 3. Application of Abel's theorem of limits. A further means of evaluating the sum of a series, one of great theoretical importance, differing from that just indicated in the principle it involves, though in most cases intimately connected with it in virtue of 101, con- sists in applying Abel's theorem of limits. Given a convergent se- ries 2 a n , the power series f(x) = 2a n x n converges at least for K x <; -f 1, and hence, by 101, 2a n = hm f(x). If we suppose that the function f(x) which the power series represents is so far known, that the latter limit can be evaluated, the summation of the series is achieved. The developments of Chapter VI offer a wide basis for this mode of procedure, and in fact Abel's theorem has already been used there more than once in the sense now explained. We shall give here only a few relatively obvious examples, with a reference to the exercises at the end of this chapter. 135* Examples. We are already acquainted with the series: 2. V-_ lim V (-!) = lim n^o 2w + 1 *->i-o,o We have the further example The series inside the bracket has for derived series 1 and therefore represents the function (v. 19, Def. 12) X dx 1 (s+1)* 1 _ l 2x-l n Accordingly, the sum ol the given series is = log 2 -f- - 4. Similarly we find (v. 19, Def. 12) For further series constructed on the same lines, the formulae of course become more and more complicated. 4. Application of the main rearrangement theorem. Equally great theoretical and practical significance attaches, in our present problem, to the application of the main rearrangement theorem. This application we proceed at once to illustrate by one of the most important cases; additional examples will again be furnished by the exercises. In 115 and 117, we obtained two entirely distinct expansions of the function xcotx, both valid at least for every sufficiently small \x\. 32. Evaluation of the sum of a series by means of a closed expression. 237 If, in the first of these, we replace x by nx, we obtain, certainly for every sufficiently small \x\, Each term of the series on the right may obviously be expanded in powers of x: * ^ v o f x \ (k -in i^vptf\ t2 "a ~~ 2j * I Taj v 2 i, *> pxea) * -x n =i N* / These are the series z (fc) of the main rearrangement theorem; since the series f (fc) of that theorem in our case only differ in sign from the series 2 (fc) themselves, the conditions of that theorem are all fulfilled, and we may sum in columns. The coefficient of x 2 ? on the right then becomes --2J^ (p fixed) and since, by 97, it has to coincide with that on the left, we obtain the important result (once more denoting the index of summation by n) n^i n*" 1 " ' ~" 71= I 'V ^ This gives us the sum of the series i__ , JL j ij. in the form of a closed expression, since the number n and the (ra- tional) Bernoulli's numbers may be regarded as known 4 . In particular, oo 1 ,8 <* - ,~4 QO i _6 V = y = y 4 Quite incidentally, formula 136 shows that Bernoulli's numbers JB. 2n are of alternating signs and that ( l)"~ 1 B in is positive; further, that they increase QO | with extreme rapidity as n increases; for since the value of rj^ lies between 1 fc=i* and 2, whatever be the value of n, we necessarily have 2(2,01 2(2)1 '~ ' whence it follows that -4- OO. Finally, as the above transformation holds for |o?|<l, it also follows that the series 115 converges absolutely at least for | x \ < n . But for | x \ > n it certainly cannot converge absolutely, for then cotcc would be continuous for x = n 9 by 98, 2, which we know is not the case; thus the series 115 has exactly the radius n. It follows from this that 116 a has the radius , 116 b the radius n. & 238 Chapter VI 1 1. Closed and numerical expressions for the sums of series. It is not superfluous to try to realise all that was needed to obtain even the first of these elegant formulae 5 . This will be seen to involve much of our investigations up to this point. The above provides us with the sum of every harmonic series with an even integral exponent; we know nothing yet of the sum of a harmonic scries with odd exponent (> 1); that is to say, we have not succeeded as yet in finding any obvious relations that might result in connecting such a sum fe. g. ^i 3 ) with any numbers occurring elsewhere. (There is of course no obstacle to our evaluating the sum of any harmonic series numerically, to any degree of approximation 8 ; v. 35). On the other hand, our results readily yield the following further formulae: We have y _ V ^n 2 * ,,t / 1 (2v The latter series is precisely the same thing as - - . Subtract- ing this from both sides, we obtain i (2 -! or For p = 1, 2, 3, . .., the sums are in particular f! ^ ^L 8 ' 96 ' 960 ' ' ' ' 1 1 If we again subtract the same series - g - ^- , we obtain v >_ */ ___ 1 1 \ 138. 1 TTr + -3r: -rrH 6 James and John Bernoulli did their utmost to sum the series The former of the two did not live to see the solution of the problem, which was found by Euler in 1736. John Bernoulli, to whom it became known soon after, wrote in this connection (Werke, Vol. 4, p. 22): Atque ita satisfactum est ardenti desiderio Fratris mei, qui agnoscens summae huius pervestigationem difficihorem quam qms putavent, ingenue fassus est omnem suam industriam fuisse elusam . . . Utinam Prater superstes esset ' A second proof, of a quite different kind, will be found in 156, a third in 189, and a fourth in 210. CO I 6 T. J. Stieltjes (Tables des valeurs des sommes S k = k , Acta mathe- n~i n matica, Vol. 10, p. 299, 1887) evaluated the sums of these series, up to the ex- ponent 70, to 32 places of decimals. 32. Evaluation of the sum of a series by means ot a closed expression. 239 In particular, fur p 1, 2, 3, ... the sums are JL a JL 4 31 o 12* ' 720* ' 30240* '"" Here again, however, we know nothing of the corresponding series with odd exponents. The last two results might of course also have been obtained by starting with the expansions in partial fractions of the functions tan or -r- , and reasoning as above for that of the func- sm & tion cot. We may deduce further results by treating the expansion in partial fractions, given in 118, of the function - r -, i.e. 1 Q K oo/1' jt 1 O | O i ^ { 1 4 cos - - 4 The r th term is here expressible by the power series V ' to (2v+l) 2 * +1 ' after rearranging, the coefficient of a; 2 " thus becomes: OO / 1 \ V 1 1 v (- 1 ) ._ 1 _ l i l i Let us denote these sums provisionally by <\ >p + 1 ; then sta; or A|~ i (*i\*+. ^YJ- 1 a [i'T~ a *( a ) ~^ ^\ n ) -i J On the other hand, this power series may be obtained by direct division and its coefficients just like Bernoulli's numbers in 105, 5 by simple recurring formulae. We usually write 77 } n 2W ~3 so that This gives E Q = 1, and, for every n I> 1, recurring formulae 7 which may be written as follows (after multiplication by (2n)\): 139. 7 The numbers determined by these formulae (which are moreover rational integral numbers) are usually referred to as Euler's numbers. The numbers Ev up to v - 30 have been calculated by W. Scherk, Mathem. Abh., Berlin 1825. 240 Chapter VIII. Closed and numerical expressions for the sums of series. or in the shorter symbolical form (cf. 106) : now holding for every k I> 1 . We deduce without difficulty: 7T 7T p _ ... _ '-i -^3 ^:> u and =1, 3 = 1, 4 = 5, fl = 61, 8 In terms of these numbers, which we are perfectly justified in con sidering as known, we have, finally, In particular, for = 0,1,2,3,..., this gives the values * * 3 5 s 61 7 4 ' 32^' 1536 ' 245 *" for the sums of the corresponding series. 33. Transformation of series. In the preceding section ( 32), we became acquainted with the most important types of series which can be summed by means of a closed expression either in the stricter or in the wider sense of the term. In the evaluations last made, which are really of a profound nature, the main rearrangement theorem played an essential part; indeed, in virtue of this theorem, the original series was changed, so to speak, into a completely different series which then yielded further informa- tion. We were therefore principally concerned with a special trans- formation of series*. Such transformations are frequently of the greatest use, and indeed even more so in the numerical calculations which form the subject of the following two sections, than in the determina- tion of closed expressions for the sums of series. To these trans- formations we will now turn our attention, and we start at once with a more general conception of the transformation deduced from the main rearrangement theorem and repeatedly applied to advantage already in the preceding section. 8 Such transformations were first indicated by /. Stirling (Methodus diffe- rentialis, London 1730); they are based, in his case, on similar lines to the above, excepting that he fails to verify the fulfilment of the conditions under which the processes are valid. S3 Transformation of series. 241 00 Given a convergent series z (k) , let each of its terms be *=o expressed, in any manner, (e. g. by 32, p. 232) as the sum of an infinite series: We shall assume further that the vertical columns in this array them- selves constitute convergent series, and denote their sums by 00 s (0) , s (1) , . . . , s (n \ .... Under what conditions may the series 2 ^ n=0 formed by these numbers be expected to converge, with jfc=0 n-O If this equality is justified, we have certainly effected a trans- formation of the given series. The main rearrangement theorem im- mediately gives the Theorem. // the horizontal rows of the array (A) all constitute 141. absolutely convergent series and denoting by f (fc) the sum, \ a n ^ |, of n=0 the absolute values of the terms in one row , if the series 2^ is convergent, the series 2 s (n) also converges and 2 2 (fe) . It is this theorem that we have applied in the preceding para- graph. The question arises whether its requirements are not un- necessarily stringent, whether the transformation is not allowed under very much wider conditions. A. In this direction, an extremely far-reaching theorem was proved by A.Markoff*. He assumes first only that the series constituted by the vertical columns of the array (A) converge, as well as the original series and the series constituted by the horizontal rows of the array. The GO OD numbers s (n) have thus determinate values. Since z* and a^ *=0 Jfc=0 converge, so does J(2 (k) a (fc) ); and also, similarly, for any fixed m, A=0 the series 2 (*<" - 4" - a? ----- 4LO (m fixed). 9 Mtfmoire sur la transformation de series (Me*m. de 1'Acad. Imp. de St. PStersburg, (7) Vol. 37. 1891). Cf. a note by the author, "Einige Bemer- kungen zur Kummerschcn und Markoffschen Reihentransformation", Sitzungs- berichte der Berl. Math. Ges., Vol. 19. pp. 417, 1919. 242 Chapter VIII. Closed and numerical expressions for the sums of series. The terms of this series are, however, precisely the remainders, each with the initial 10 index m, of the series constituted by the individual rows of the array. If, for brevity, we denote these remainders by r^\ so that r = 2 a (k and m fixed), rt=OT the series 2 r ( * } - R m (m fixed) is convergent. The further assumption is then made that R m -> when m -> oo. It may be shown that under these hypotheses 2sW converges and = The theorem obtained will thus be as follows: 142. Markoff's transformation of series. Let a convergent series 00 2 zW be given with each of its terms itself expressed as a convergent series: k--o (A) *W = V*) + i w + + *n w + - (* = 0, 1, 2, . . .). QO Let the individual columns 2 a n W of the array (A) so formed represent A=O convergent series with sum $(">, n = 0, 1, 2, . . . , so that the remainders of the series in the horizontal rows also constitute a convergent series 00 k ^Jm = R m (m fixed). In order that the sums by vertical columns should form a convergent series Z s("\ it is necessary and sufficient that lim/^ m = R should exist; and in order that the relation 2 s< fl > = 2 zW n = A=0 should hold as well, it is necessary and sufficient that this limit R should be 0. The proof is almost trivial, for we have (a) > + sW +... + ,00 = R - R n+1 , whence the first statement is immediate. Since it follows that n=0 M = R Q -R, (to and since R is simply 2 r n ) = 2 z^ k \ the second statement now follows also. " * =0 10 Here we of course take m to give the whole series, i. e. z (k) itself. 33. Transformation of series. 243 B. The superiority of Markoff's transformation over Theorem 141 consists, of course, in the absence of any mention of absolute conver- gence, only convergence pure and simple being required throughout. Its applications are numerous and fruitful: those bearing on numerical evalua- tions will be considered in 35, and we shall only indicate in this place one of the prettiest of its applications, which consists in obtaining a trans- formation given by Euler u of course, in his case, without any con- siderations of convergence. It is advantageous here to use the notation of the calculus of finite differences, and this we will accordingly first elucidate in brief. Given any sequence (# , x lt x 2 , .)> tne numbers are called the first differences of (x n ) and are denoted by A # , A x l9 . . . , A x k9 . . . The differences of the first order of (A x n ) 9 i. e. the numbers A x k A x kl , lt k = 0, 1, 2, . . . , are called the second differences of (x n ), denoted by J 2 * , A*x l9 ..., A*x k , . . In general, we write for n ^ 1 J+* * t = J * fc - J- * &+1 (A = 0, 1, 2, . . .) and this formula may also be taken to comprise the case n if we in- terpret A" x k as being the number # A itself. It is convenient to imagine the numbers x k and A n x k arranged in rows so as to form the following tri- angular array, in which each difference occupies the place in its own row immediately below the space, in the row above, between the two terms whose difference it is: Q, X}, % 9 #3, Ax Qj Ax lt Ax 2J (A) A*x , The difference A" x k may be expressed in terms of the given numbers x k directly. In fact J 2 x k = A x k - A X M = fa - X M ) - x k 2 X k+l -f- and similarly 3 x k+2 11 Institutiones calculi diffcrentiahs, 1755, p. 281. 244 Chapter VIII. Closed and numerical expressions for the sums of series. 143. the formula A Xk = x k - (J)* m + (J)* M - + ... + (-D" (J) *<-+ for fixed &, is thus established in the cases n = 1, 2, 3. By induction, its validity for every n follows. For, supposing 143 proved for a particular positive integer n, we have for n + 1 : whence by addition, since (j + ( ^j) = ( \ we have the formula 143 for n + 1 instead of w. This proves all that is required. Making use of the above simple facts and notation, we may now state the following theorem: 144. Euler's transformation of series. Given an arbitrary con- vergent series 12 QO we invariably have: t i \ 7c ^P ^ ^o k n-0 2 n+1 ' i. e. the series on the right also converges and has the same sum as the given 12 The series need not be an alternating series, i. e. the numbers a n need not all be positive. There are however small, though by no means essential, advan- tages in writing the series in alternating form as above, when effecting the trans- formation. 13 This general transformation is due to Ruler (Inst. calc. diff., pp. 281 seq., 1755). The particular transformation given below in example 2 is to be found already in a letter to Leibniz dated 2. 8. 1704, from/. Bernoulli, who attributed the discovery to N. Fatzius. (Cf. also J. Hermann, letter to Leibniz of 21. 1. 1705.) An early investigation of a more searching kind, using remainder terms, was under- taken by /. V. Poncelet, Journ. f. d. reine u. angew. Math., Vol. 13, pp. 1 seq., 1835. The proof that the transformation is always valid, provided only the series 2 ( l) fc a k is assumed also convergent, was first given by L. D. Ames (Annals of Math., (2) Vol. 3, p. 185. 1901). Cf. also E. Jacobsthal (Mathem. Zeitschr., Vol. 6, p. 100. 1920) and the note bearing on that by the author (ibid. p. 118). 33. Transformation of series. 245 Proof. In the array (A) of p. 211, we substitute for a n (k) : (b) a^ = ( - 1)* [ 2 L n A"a k - ^ A- a J . By 131, if we now sum for every n y keeping k fixed (i. e. form the sum of the k lh horizontal row), we obtain QO j For fn\ fn\ . , / i f n \ lim A" "* lim W**" U/**' i+ " ' ' " ' ( > W*" + * is equal to zero by 44, 8, because a ky rt /<,fi> #/cf-2> certainly form a null sequence. Accordingly (b) gives an expression for the individual terms of the given series S ( I) 1 &L in infinite series. Forming the sum of the th column, we obtain the series Z(-l) k Ln A n a k ~ - 2 n + i J n+1 <i, 1 (n fixed) ; - J the generic term of this series, as J n+1 a k A n a k A n a k+1 , can be written in the form = IV. K - 1 ? A" a k - (-!)*+ J" A+] ], so that the series under consideration may again be summed directly, by 131. We obtain % n (ft) = nVi t jn tf o - Hm ( ~ l) fc J n ^] ( fixed). k^O k-^co Since, however, the numbers a L form a null sequence, so do the first differ- ences and the th differences generally, for any fixed w. The vertical columns are thus seen to constitute convergent series of sums The validity of Euler's transformation will accordingly be established when we have shown th; to have the values we have shown that R m -> 0. Now the horizontal remainders are seen (G51) 246 Chapter VIII. Closed and numerical expressions for the sums ol series. following precisely the bame line of argument as was used above for the entire horizontal rows. Thus *-=^J;(- i )^"* ( fixed **) If we write for brevity this series for R m may be thought of as obtained by term-by-term addition from the (m -f- 1) series: Hence therefore, as r w is the term of a null sequence, so is /? w> by 44, 8. This proves the validity of Etilers transformation wilh full generality. Examples. 1. Take ..! + +_.... 234 The triangular array (A) takes the form i I * I I 1, 2 -, 4 5 1111 F2' 2T3 J 3 4' 4-5' '' 1 2 J_-2 1 2 i.2~3' 2T3-4 ' 3 "4 5' "" 1-2.3 1-2.3 1-2 3-4' 2-3-4.5' ' " The general expression of the n th difference is found to be 4 . a , _ Ml (* + l* so that in particular This is easily verified by induction. Accordingly we have tOi 1 ! 11 ! ! I * I * I * I 8 = lOg 2 = 1 + -5- -r H -- .-.= - r -^ -- ^ H -- - -\ -- j -f . 3 8 4 ' 1.2 1 2-2 :! 3-2 a 4-2* The significance of this transformation e. g. for purposes of numerical calcu- lation ( 34) is at once apparent. 2. With equal facility, we may deduce * 1t 1 I11<a1 ' a ' i In what cases this transformation is particularly advantageous for pur- poses of numerical calculation will be seen in the following section. 34. Numerical evaluations. 247 C. Kummer's transformation of series. Another very obvious transformation consists simply in subtracting from a given series one whose sum is capable of representation by means of a known closed expression and which at the same time has terms as similar in con- struction as possible to those of the given series. By this means, subtracting for instance fiom s = 2,'- t2 the known series (v. 68, 2 b) !_ v_.JL_ - (+!)' we deduce the transformation CO 1 QO 1 The advantage of this transformation for numerical purposes is at once clear. Simple and obvious as this transformation is, it yet forms what is really the kernel of Rummer's transformation of series 1 *; the only difference being that a particular emphasis is now laid on a suitable choice of the series to be subtracted. This choice is regulated as follows: Let 2a n = s be the given series (of course, by hypothesis, convergent). Let Sc n = C be a convergent scries of known sum C. Let us suppose that the terms of the two series are asymptotically proportional, say lira -.= = 7 4= . n-> c >* In that case ,45. and the new series occuring on the right may be regarded as a trans- formation of the given series. The advantage of this transformation lies mainly in the fact that the new scries has terms less in absolute value than those of the given series, as in fact (l y ] >Q. Con- sequently its field of application belongs for the most part to the do main of numerical calculations and examples illustrating it will be found in the following paragraph. 34. Numerical evaluations. 1. General considerations. As repeatedly explained already, it is only on very rare occasions that a closed expression, properly so- called, exists for the sum of a series. In the general case, the real 11 Kummer. E. E.: Journ. f. d. reine u. angew. Math., Vol. 16, p. 206. 1837- Cf. also Leclert and Catalan, Memoircs couronne's et de savants etrangers de 1'Ac. Belgique, Vol. 33, 1865 67, and the note by the author mentioned in footnote 9. 248 Chapter VTII. Closed and numerical expressions for the sums of series. number to which a given convergent series, or the sequence of num- bers for which it stands, converges, is, so to speak, first defined (given, determined, . . .) by the series itself, in the only sense in which a number can be given, according to the discussion of Chapters I and II 15 . In this sense, we may boldly affirm that the convergent series is the number to which its partial sums converge. But for most practical purposes we gain very little by this assertion. In practice, we usually require to know something more precise about the magnitude of the number and to compare different numbers among themselves, etc. For this purpose, we require to be able to reduce all numbers, defined by any kind of limiting process, to one and the same typical form. The form of a decimal fraction is that most familiar to us to-day, and the expression, in this form, of numbers represented by series accor- dingly interests us first and foremost 10 . The student should, however, get it quite clear in his own mind that by obtaining such an expres- sion we have merely, at bottom, substituted for the definition of a number by a given limiting process, a representation by means of another limiting process. The advantages of the latter, namely of the decimal form, are mainly that numbers so represented are easily com- pared with one another and that the error involved in terminating an infinite decimal at any given place is easily evaluated. Opposed to this there are, however, considerable disadvantages: the complete ob- scurity of the mode of succession of the digits in by far the greater number of cases and the consequent labour involved in their succes- sive evaluation. The sc advantages and disadvantages may be conveniently illustrated by the two following examples: (l -) 1 -i+] -y +-- = 0-785398. .. l-i+i--H ----- = 0-693147... By the series, distinct laws of formation are given; but they afford us no means of recognizing which of the two numbers is the larger of the two, for instance, or what is its excess over the smaller number. The decimal fractions, on the other hand, exhibit no such laws, but give us a direct sense of the relative and absolute magnitudes of both numbers. 15 Indeed an infinite series our previous considerations give ample confirmation of the fact is one of the most useful modes of so defining- a number, one of the most significant both for theoretical and practical purposes. ld And only in special cases the expression in ordinary fractional form. The reason is always that of convenience of comparison; which, of -J^ or |$, is the larger, we cannot say at once, whereas the answer to the same question for 647 and 0-641 requires no calculation whatever. 34. Numerical evaluations. 249 We shall therefore henceforth reserve the term numerical eva- luation for the expression of a number in decimal form. As no infinite decimal fraction can be specified in toto, it will be necessary to break it off after a definite number of digits. We have still a few words to say as to the significance of this process of breaking off decimal fractions. If it be desired, for instance, to indicate the number e by a two-digit decimal fraction, we may with equal justification write 2' 71 and 2*72, the former, because the two first decimals are actually 7 and 1, the latter, because it appears to involve a lesser error. We shall therefore make the following con- vention: when the n specified digits after the decimal point are the actual first n digits of the complete infinite decimal which expresses a given number, we shall insert a few dots after the w th digit, writing for instance a = 2*71...; when, however, the number is indicated by the nearest possible decimal fraction of n digits, we insert no dots after the w th digit, but write 17 e. g. e & 2-72, in the latter case the ;i th digit written down is thus the w th digit of the actual infinite fraction raised or not by unity according as the succeeding part of the infinite fraction re- presents more or less than one half of a unit in the w th decimal place. In point of fact, cither specification has the effect of assigning an interval of length l/10 n containing the required number. In the one case, the left hand end point is indicated, in the other, the centre of the interval. The margin, for the actual value, is the same in both cases. On the other hand, the error attaching to the indicated value, relatively to the true value of the number considered, is in the former case only known to be I> and _? ]/10 n , in the latter to have modulus 5^ i/10 n . We may therefore describe the first indication as theoretically the clearer, and the second as practically the more useful. The diffi- culty of actual determination of the digits is also in all essential par- ticulars the same in both cases. For in either, it may become ne- cessary, when a specially unfavourable case is considered, to diminish the error of calculation to very appreciably less than 1/10 w before the n th digit can be properly determined. If we are, for instance, concerned with a number cc = 5-27999999326 ..., to determine whether a = 5'27 ... or 5*28 . . . (retaining two decimals), we have to diminish the error to less than a unit in the 8 th decimal place. On the other hand, if we are concerned with a number /? = 2'3850000026 . . . , the choice between /?^2'38 and 2*39 would be influenced by an uncertainty of one unit in the 8 th decimal place 18 . 17 In e = 2-71..., the sign of equality may be justified as representing a limiting- relation. 18 The probability of such cases occurring- is of course extremely small. By mentioning them, we have merely wished to draw attention to the signi- ficance of these facts. In Kx. 131, however, a particularly crude case is indicated. 250 Chapter VIII. Closed and numerical expressions for the sums of series. 2. Evaluation of errors and remainders. When given a conver- gent series a n =s, we shall of course assume that the individual terms of the series are "known", i. e. that their expressions in decimal form can v easily be obtained to any number of digits. By addition, every partial sum s n may accordingly also be evaluated. The question 19 remains: what is the magnitude of the error attaching to a given s n ? Here the word error designates the (positive or negative) number which has to be added to s w to obtain the required value s. Since this error is s s n , i. e. is equal to the remainder of the series, starting im- mediately after the n th term, we will denote it by r n , and the process of determining this error will also be designated by the term evaluation of remainders. In practical problems, evaluations of remainders almost invariably reduce to one of the two following types: A. Remainders of absolutely convergent series. If s = 2 a n con- verges absolutely, determine a scries JE a n ' of positive terms, capable of summation in a convenient closed expression, and with terms not less than the absolute values of the corresponding terms of the given series (though also exceeding these by as little as possible). Obviously which is assumed known, thus provides a means of estimating the magnitude of the remainder r n , i. e. \r n \<^r n ', and this all the more closely the less a n ' exceeds \a n \. A particularly frequent case is that in which, for some fixed m, and every k ^> 1: k,, ffc |^|* m -a" with 0<a<l; in that case, ot course, and in particular, if < a <^ |: KJ^KJ. The absolute value of the remainder is in this case not greater than that of the term last calculated". B. Remainders of alternating series. Given a series of the form s = 2( l) n # w and supposing that the (positive) numbers a n form a monotone (decreasing) null sequence, we have (cf. 82, Theorem 5): < _ ln + i r _ fl _ * + _ 19 Or in more practical form: Up to what order of decimal does s w coin- cide with the required value s? 80 In forming- these estimates, it should be noticed that they give no in- dications as to the sign of the remainder r nt only as to its absolute value. 34. Numerical evaluations. 251 Hence we may assert that the error r n has the same sign as the first neglected term, but has a smaller absolute value. When neither of these two modes of procedure is applicable, the evaluation of remainders is usually more troublesome, and it becomes necessary to adopt special artifices in each particular case. We shall, then, designate the series considered as rapidly or slowly convergent, according as r n does or does not fall within the desired limit of error for moderate values 21 of n. A few further fundamental remarks may be elucidated by the 3. Evaluation of the number e. We found ^ == iiJ_i_L_fJ__i. |_ JL _L . . . . e A ^l!^2! h 3!^ ^ wM Already, on p. 194, we have mentioned that the (positive) remainder r n was less than the n th part of the term immediately before, so that s << s - + sn;- In effecting the numerical calculations, we have now to take into ac- count the following fact: When we express the individual terms of the series in decimal form, we have even at that point to break off the decimals at some particular digit, and we therefore incur a certain error. Unless n remains comparatively small, these errors may accu- mulate to such an extent that the whole calculation is in danger of becoming illusory. The mode of procedure is then as follows: Sup- posing that we are retaining 9 digits, we write 22 a + a^ -|- a = = 2-500000000 a { ~ = 166 666 667" 4 =--0-. 41 666667" 0, =0-.. 8 333333+ a (} =- 0*.. 1388 889~ a 1 0-... 198413- 8 =-0-.. .24802- a 9 =-0- 2756" 10 =0- 276- a^ --() 25 + =0- 2+ ['i, <0' 0+] Here the small -f- an( ^ signs are intended to indicate whether the error in the term in question is positive or negative. In either case it is in absolute value less than one half of a unit in the last decimal place. By addition, we obtain the number 2-718281830. 21 A more precise definition of rapid convergence will be given in 37. 82 a n is deduced from <*_! by simple division by n. 252 Chapter VIII. Closed and numerical expressions tor the sums of series. But s 13 itself may possibly (namely if all positive errors are nearly and all negative ones nearly } 2 of a unit in the last decimal place) fall short of the number required by as much as | of a unit in the last decimal place; or it may, on the other hand, be as much as jj of a unit in excess, since there are 7 negative and 3 positive errors. Taking also into account the remainder, we can only deduce with certainty, since s n < e = s n + r n , that 2-718281826 < e < 2-718281832 . Our calculation thus secures only the first seven true decimals, while the approximate value 23 is obtained with eight digits: e f^ 2-71828183. In practice it will generally suffice to proceed a few decimal places further (2 or 3 at most) with the evaluation of the terms than it is desired to proceed for the sum. The number n of terms taken into account will be chosen so large that the remainder r n contributes at most one unit in the last decimal place considered. The error in the individual terms will then, in general, have no appreciable effect. But to obtain perfect security for the resulting- digits, it is necessary to proceed as described above. For we may retain a large number of digits beyond the desired number in calculating the individual terms, yet as an error attaches to each of the decimals broken off and these errors accumulate, they may, in particularly unfavourable cases (cf. the example 01? p. 249), influence some of the much earlier digits 4. Evaluation of the number yt. The chief means placed at our disposal, up to the present, for the evaluation of the number n, are the series expansions of the functions tan"" 1 and sin" 1 ; of these, the former has the preference, owing to its simple mode of formation. From this series, we deduced the expansion T w=sl ~~T + T -- 1 ' which for numerical purposes is practically valueless. In fact, by p. 250, we can say no more on inspection about the remainder r n in this expansion, than that it has the sign ( l) n+1 and is in absolute value < - - . In order to secure 6 decimals, we should therefore & n -f- o be obliged to take n > 10 fl , but an evaluation of a million terms is, for practical purposes, quite impossible. The rapidity of the conver- gence may be increased very materially by Euler's transformation 144, 2. In the next paragraph, we shall discuss the utility of such transformations for purposes of numerical calculation. Our present object is to deduce more convenient series expressions for n directly from the tan"" 1 series itself. The series expansion for tan"" 1 = ~ is already of appreciable \/ 3 6 use: this gives fL^j_r 1 __i_ + _ 1 ___ L _+_... i 6~~V~5l 33^5-3* 7-3 3 ^ J' 23 Cf. p. 249. 34. Numerical evaluations. 253 The following mode of procedure, however, provides considerably more convenient series 94 . The number a = tan * y ^ y ~ 3.53 + 5 . 5 ^ y. 5 7 H is easily calculated from the series itself (see below). For this value of a, tan a = | , and so _ 2 tan a 5 tan 2 a = - r- =^= TT> 1 tan- a 1& and 120 Consequently 4 a exceeds -*- by only a small amount. Writing *- J=/, we have t ^ __ _ism 4_o_--jan * ^L. _ JL ^ ~ 1-f tan 4 a tan i"w ~~~ 239' Hence ft can very easily be evaluated from the series ~ _, _1 __ JL __ I I , p tan 23g - 2 g g 3- 239 3 ^ ' The two numbers a and fi give us 146 ' If it be desired to obtain the first seven true decimals of n , we may endeavour to attain this end by taking-, say, 9 decimals for each of the terms and for the remainder' 26 a scanty enough margin, for the errors incurred on the numbers a and ft have ultimately to be multiplied by 16 and 4 respec- tively Denoting- the first series by ^i ^ + ^ft -- [-> lne second by a/ a 8 ' -{- a 6 ' [- an< * tne corresponding partial sums by s y and s v ', the calculation proceeds as follows: <*! - 0200000000 a, - OOOOOG4000 0- T>7 - a L -}- a, -f- 07- 0^00004057- = 0002G66(>G7- -- 0000001829- - 0- 2- ~|- a, + u - 002608498 Hence, as the errors change signs in a subtraction, Sll = 0- 197 395 559 + + + - and < fn < 10 10 Accordingly 3 158328936 < 1ft a < 3-158328970, 24 J. Machin (in W.Jones: Synopsis, London 1700). 26 The result alone can show whether this suffices. In fact we do not know a priori whether we are not in the presence of one of the particularly unfavourable cases described on p. 249. 9 (G51) 254 Chapter VIIT. Closed and numerical expressions for the sums of series. for after multiplying by 16 we have to subtract ^=8 units of the 9th deci- mal place, or add = 24 of these units, to obtain bounds on either side for 16s n . Since we have finally to add 2 units to the bound above, to obtain the correspond- ing bounds of 16 a. Further < = 0004 184 100 + */- a' = 0004 184 076* hence - 016736307 < - 4 ft < - 0-016736302 . Combining the two results, we get 3-141 592629 < n < 3*141 592668 . This brief calculation thus really gives us the seven first true decimals of n\ * = 3-141 5926 .. (The same procedure would only have secured six decimals for the approxi- mate value; cf. calculation of 0, where circumstances, in this respect, were the exact reverse) The series here utilized for the calculation of n are among the most convenient; by their means, a very much greater number of decimals may also be secured 28 with relatively small trouble, and we are therefore fully justified in regarding n henceforth as one of the ''known" numbers. 147. 5. Calculation of logarithms. The starting point for the cal- culation of logarithms resides in the series This series converges with considerable rapidity for x = \, and at once gives Denoting by , 1 ,..., the terms of the series inside the square bracket, we have \ and . . . i [ill i A ....] i i < 0<r or 1 1 9 (2n + l) 3- >w + l Z* 8 ~~ 8 ' 26 The number n has been evaluated to 810 places of decimals (Mathematical Gazette, Feb. 1948, p. 37). 34 Numerical evaluations. 255 Our calculations then proceed as follows, if we again take 9 decimals for each of the terms a n - a = 0333 333 333+ ! 001 2 345 679+ a, -0000 823 045+ ^-0-... 065 321+ 4 :=()... 00:> 645+ fl fi -0- ...... 513+ -0- ..... 048+ a, = 0- ..... 005- 0346573589 Whence it follows, taking into account the remainder and the small -+- and signs: log 2 = 0-693 147 1 ... or log 2 ^ 0-693 147 2 with seven decimals secured 27 . Once log 2 is evaluated, the calculation of the logarithms of all other numbers involves very little further trouble. In fact, our senes gives, for * = -, ; 148. ' therefore if log /> is known (/> = 2, 3, . . .), we obtain the value of log (p -(- l), by the above formula. Moreover, since ^ - r = , -->> the expression involves a series converging very rapidly. In fact (cf. above, case p = 1) so that the remainder is already very small for quite moderate values of n. The rapidity of convergence of course increases when p is given somewhat larger values, i. e. as soon as the first few logarithms have been successfully determined. It is useful to observe that by 37, 1, only logarithms of prime numbers 2, 3, 5, 7, 11, 13, . . . need be evaluated; those of all other numbers follow by mere combination. Now supposing that we have effected the calculations for the logarithms of the first four prime numbers, 2, 3, 5, 7, the labour in. volved in calculating the logarithms of further primes is small. Thus, for instance, taking p = 10, we have with 11-40' 27 The series 1 J-f-J }+ for logf 2 is of course inappropriate for the evaluation of this number; even its Enler's transformation effected in 144, 1 is less convenient than the series utilized above. 256 Chapter V11I. Closed and numerical expressions for the sums of series Thus already for n = 3, r < -1 < - *- < - l < *- * ^ 7-21 7 11-40 ^ 20 R -2-ll-7 ^ 10-2 D -7 10 ia ensuring a degree of approximation sufficient even for the most refined scientific needs. It would accordingly appear desirable to possess somewhat more con- venient methods of calculation for log- 2, log 3, log 5, and also, at any rate, log 7. Diverse artifices may be applied for the purpose, all of which consist k mainly in finding rational numbers , as near as possible to 1, whose m numerators and denominators are products of powers of these first four primes. If q of these primes have been utilized, q fractions will be needed to deduce the logarithms of those q primes from those of the fractions. For actually effecting these calculations, it is convenient to follow the method indicated by Adams**: Evaluate the logarithms of - , ^r, ^- by means, not of the series 7 <JTC Ov/ 12O, c just employed, but of the original series 12O, a and b, which here give 25 . / 4 N 4 1 16 1 64 81 / 1 \ 1 1 1 1 Owing to the occurrence, in the denominator, of powers of 10, the calculation here becomes extremely simple With the aid of these logarithms, we then obtain, as may be verified immediately: 1og2= 71og- - lo K 3 = 11 log--- Slog 2 + 5 log M 10 2^ m logs -mo* -4 to* "J+TidarjjJ. It we proceed further to evaluate, as we may with equal facility, 39 . 126 / 8 \ 8 1 8 3 1 8 3 IOff ~ e + "^~"^ + "^~" f "" f we also obtain ,n, 10 ., 25 01 81 , 12(5 * 8 Proc. of the Royal Society, Vol. 27, p. 88, 1878. 29 The facility with which this calculation is effected may be seen by 126 the following, which in 5 simple lines provides log -^~ with 10 decimals secured: -f 0-008000000000 -0 ...032000000 4-0- 170667- -0- 001024 4-0- 007- logA- = 0-0079681696.., 34. Numerical evaluations. 257 We have thus, for the actual calculation of natural logarithms, a method which is convenient and easily applicable in practice. Into further details of the com- putation of logarithmic tables we cannot enter in this place. Having obtained log 2 and log 5, we have also the value of log 10; and hence, in M= I = 0-434 294 48190 .., log 10 f the "modulus" of Briggs' system of logarithms to the base 10, or factor by which the natural logarithm of a number must be multi- plied to give the Briggian logarithm 30 . 6. Calculation of roots. Once logarithms have been mastered no great practical importance attaches to the problem of obtaining simple methods of calculation for the roots of natural numbers. We shall therefore be quite brief in the following explanations. The ra- pidity of convergence of the binomial series increases as | x \ diminishes. Now the calculation of a power Vq = q p i can always be reduced to that of a power of the form (l + #) p , with some small value of \x\. A few examples may serve to illustrate the above. On p. 211, we gave 149. for ^/2 the series expansion of FM"?^) : ^. 1 + 3 4-' 5 . +. 5 "*" 2 50"*" 2-4 50' jn ~ 2-4-6 50 ^ Since ( l) n I *J is constantly positive and forms a monotone decreasing quence, the remainder r n may be estimated by means of the inequality showing that, even for small values of n, a considerable degree of approxi mation is attained* 11 . The method is even more effective if we write 30 We may remark in passing that we have certainly found ample justi- fication, by this time, for what seemed at first the rather arbitrary designation of the logarithms with the remarkable base e as the "natural" logarithms. 31 How simply the calculation proceeds is shewn by the following details: - ^ = 1 010 ......... hence indeed without any error! 1-0101525445375 a 2 = 0-... 15 3 = 0- 25 a4 = n' 43 ^;!? \/2"= 1-4142135623..., Clft U* i o / D by which the first 10 decimals are thus already seemed. 258 Chapter VIII. Closed and numerical expressions for the sums of series. or other similar expressions, obtained by taking* any rough approximation a to ^2 f~ in the first case, 1-41 in the second J, and putting Since a 2 is chosen to be very near 2, the quantity under the ^/~ is of the form 1 -\- x t with small \x\. Similarly, if we are already aware that y/3 = 1*732 . .., we have only to write to obtain, with the greatest ease, an expansion of y/3 to 50 or more places ol decimals. We may, without further explanation, indicate the examples: 150. 7. Calculation of trigonometrical functions. The series expansions of sinx and cos a; converge with even greater rapidity than the ex- ponential series, since only the even or only the odd powers occur in them, and these have, moreover, alternating signs. Accordingly, no special artifices are required; for angles of no excessive magnitude, the series furnish all that can possibly be desired. To determine, for instance, sml, we have first to express 1 in circular measure. We have 1 = -^r = 0-017453292 . . . , i. e. ccr- lou tainly < p- - . Denoting this quantity by a, ou sin 1 = a ~ + j -- 1 ---- = a - a + a 2 -- 1 and the error r n may at once be estimated (p. 250, B) by which last expression is already less than ^--lO" 15 f r n = 2. Circumstances are similar in the case of cos 1; this quantity may also, however, since sin 2 1 < 5^, be obtained easily from the relation ZoUU cos 1 = (1 - sin 2 1)* by means of the binomial series: tana; and cot a; are then obtained by division, or from their expansions 116 and 115, whose conver- gence is still quite sufficiently rapid when |a:| is small. These latter series also lead to useful expansions for the log- arithms of sino; and cosjc, which for practical purposes are of g 34. Numerical evaluations. 259 greater importance than the values of sin x and cos x themselves. We have 32 (cf. 19, Dcf. 12) log sin x = log x -f- log ^-^ = log x -J- I cot x -- 1 dx 151. and similarly from 116 r - log cos *= (tanxdx = j?(-l)*- 1 ^4^"^^; J J=l <S/?-(//?J' log tan a; and log cot a; may be obtained from these by simple addition. As regards the convergence of these series, we can only state in the first instance that they certainly do converge for all sufficiently small values of \x\. However, the remarks of p. 237, footnote 4, show further that the series in 151 has the radius n y that in 152 the radius ^. u Further details in the computation of trigonometrical tables will not be entered into here, as they do not concern the theory of in- finite series. 8. More accurate evaluation of remainders. In the cases pre- viously considcied, the sum of a given series was invariably deduced by evaluating suitable partial sums and estimating the error involved in the corresponding remainder. It is obvious that this method is im- practicable unless the convergence of the series is relatively rapid. If it be desired to evaluate, with some degree of approximation, for instance 00 1 this direct method is pretty hopeless 33 . Even if we are very cautious in the margin we allow, we can only deduce, as an upper estimate of the remainder 32 The function in the square bracket has to be understood to stand for the series 115 after division by x and subtraction of the foremost term . x The function is therefore defined and continuous also for # = 0. 33 As we happen to know that the sum is , its evaluation indirectly by means of the value of JT is of course quite simple. But for the moment we are assuming that we know as little about this sum as e. g. about the sum 260 Chapter VIII. Closed and numerical expressions lor the sums of series. the inequality __ ____- n (n + 1) ~~ (n + 1) (n + 2) n ' according to this, it would become necessary to calculate a million terms, in order to secure 6 places of decimals. This of course is out of the question. This state of things may frequently be improved to some extent, if it is possible to supplement the upper estimate of the remainder r n by a lower estimate, i. e. to deduce an inequality for r n of opposite sense to the above in our case. In our example, the same principle as that already used gives 1 1 _____ 1 l~ " * " we are thus able to assert that our sum s satisfies the conditions for every n. To secure 6 decimals, we may accordingly need only 1000 terms. This is still too large a number for practical purposes. But in special examples this method of upper and lower estimates of the remainder (cf. Ex. 131) may lead to a satisfactory result. These cases are, however, so rare, that they do not come into account for practical purposes. Greater importance attaches to methods for transformation of slowly convergent into rapidly convergent series, because they admit of a far wider range of applications. To these methods we proceed to give our attention. 35. Applications of the transformation of series to numerical evaluations. In cases of slow convergence, one naturally attempts to change the given series into one with a more rapid convergence, by means of some suitable modification. We proceed to examine in this light the transformations discussed in 33, so as to see how far they will be of use to us here. A. Kumtner's transformation. For this transformation it is im- mediately obvious whether and to what extent an increase in the ra- pidity of the convergence can be obtained by it. In fact, using the notation of 145, we have n=0 n=0 as (1 r- j *0, the terms of the new series (from some index on- \ (t / wards) are less than those of the given series. The method will ac- 35. Applications of the transformation of series to numerical evaluations. 261 cordingly be all the more effective the smaller the factors ( 1 y ) \ a-n/ are, from the first; or in other words, the nearer the terms of 2c n are to those of 2 a n - Examples. 153 1. We found on p. 247 that J^ = 1 + V__ - -- . The terms of the *-' n* ** n* (n -f- 1) new series are asymptotically equal to those of the series oo I 1 *, / 1 1 \ 1 + 2) = Y ( + l) ~ (+l)( + 2) = "4 thus here C r- and y = l t and so Proceeding 1 in this manner, we obtain, at the />th stage: 1.1. . i . ,. V __ ^ n 2 ~ The latter series, even for moderate values of p, shows an appreciably rapid convergence. 2. Consider the somewhat more general series " 1 f a arbitrary -{= 0, -1, ... Here we take ^n = (w+y)a n -(w-|-l+y)fl nh l, W = 0, 1, 2 ..... and we try to determine y (independent of n) so that c n is as near a n as possible 34 . Here we have C = y a and a simple calculation gives y ~ - . ^ p 1 Hence we obtain The expression in the large bracket is - _ Since, by simplification, the terms in w 8 and w a must disappear of themselves, this gives - 1 -y) (K (2 ?-!)"( + + />) terms in n also disap then the expression in the large bracket above now becomes Q If we now choose y so that the terms in n also disappear, i. e. take y = a-l ^-/> 1 34 The choice of a number c n of the form x n x n \-\ will, by 131, always prove most convenient, as in that case C at any rate may be specified at once and the choice still be so arranged that the c n 's are near to the a n 's. 262 Chapter VJJ1. Closed and numerical expressions tor the sums of series. and accordingly 3 The transformation thus has the effect of introducing an additional quadiatic factor in the denominator. Particular cases: a) = 1. p* ~ 1*~2* . . . ^"" r 2"(2> - 1)^ (n + 1) J . . . (n + p)* (n Write for brevity i)' ~ M , OH- i) j . .T^T the result then takes the form _ _ _ p ~2(2p- 1) ."l a -"2 a . .". p a "" 2 (2 p - 1) ^ 1 ' This formula enables us easily to obtain very rapidly convergent series for b) Similarly, for a = - t y : Li This formula similarly leads to rapidly convergent series for ^S/6~^ri\a' For further examples, see Exercises 127 seq. B. Euler's transformation. Euler's transformation 144 need not by any means involve an increase in the rapidity of convergence 35 of the series to which it is applied. 06 / 1 \ n 154. For instance the transformation of (-^-) gives the series n=0 > Z ' 1 * / 3 \ n - V (--) , which evidently has a less rapid convergence. But even <2 ^ 4 ' / 35 The explicit definition of what we mean by more or less rapid con- vergence will be given in 37: 2a n ' is said to be more or less rapidly con- vergent than 2a n1 according as - or -* + oo . 35. Applications of the transformation of series to numerical evaluations. 263 in the case of alternating series, the effect need not be an increased rapidity of convergence; indeed the following three examples show that all conceivable cases may actually occur here: 00 1 1 1 1. ( *) n o^ S ives a more ra P icil y convergent series, - is- n=o J z n=u 4 00 J 2. J ( l) n ^ series with the same rapidity of conver- n=o 6 1 1 gence, -g- JJ -^ . J n=0 d 3. '( If-/* less rapidly convergent series, - - Jj (j-) - n=0 * * n=0 vo/ We shall now show, however, that such an increase in the rapi- dity of the convergence does result, in the case of those alternating series 2(\) n a n , a n > 0, whose terms, though not showing rapidity of convergence, still tend to zero in a particular regular manner, which we proceed to describe. These are the only types of alternating series of any practical importance. The hypothesis required will be that not only the numbers a n form a monotone decreasing sequence, i. e. have positive first diffe- rences A a n , but that the same is true of all differences of every order. A (positive) sequence a , fl,,0 2 , ... is said to have p-fold monotony if its first, second,..., /> th differences are all positive, and it is said to be fully monotone if all the differences J 7c a n , (k, n = 0, 1, 2, . . .) are positive. With these designations, the theorem referred to is: QO Theorem 1. // J^(~ l) n a n ^s an alternating series for which 15S n=0 the (positive] numbers , 1 ,... form a fully monotone null se quence, while, from the first, - n * * ^ a > (for every n) 37 , then the .1 a " transformed series L~^^ n a Q converges more rapidly than the given series. The proof is very simple. As M+1 ^ a, we have a n ^ # a n . a n Further, for the remainder r n of the given series we have (~l) n+1 r n = a n ^ - a n+2 + _...= A a n+1 + A a n , 3 + A a M + . , hence, since (A a v ) is itself a monotone null sequence, I \-> l fA _i_/l A_ A . \- 1 -* 1 I T<n I i-= 2 ' ^w+l ~T ^J a n \ 2 I ^ fl n+3 "T ~" 2 ^w+1 = 2 ^ 38 Cf. Memoir of E. facobsthal referred to in 144. 37 This assumption is the precise formulation of the expression used above, that the given series should not converge particularly rapidly. The series will in (1\** 27 *_i. luruicr me wur* uy r. v . runteici. quuicu in IUUIHULC *o. ' 264 Chapter VIII. Closed and numerical expressions for the sums of series. On the other hand, as /4 n a A n ' ] a A n a^ ^> 0, the numerators of the transformed series also form a monotone null sequence, and in particular are all <^ a . Consequently the remainders r n ' of the transformed series, which, moreover, is a series of positive terms, satisfy *n ~~2~*H-~a I" " " " ~ 2 n + 2 \ * 2 4 " Consequently, we have which proves our statement completely. Further, we sec that the larger a is, the greater will be the increase in the rapidity of con- r ' vergence, i. e. the more rapidly will >0. In particular, we may *n transform into series which converge with practically the same rapid- (1 \ n -g-J , all alternating series for which the ratio of two con- secutive terms tends to 1 in absolute value; such series have usually a slow convergence. Examples. The two most striking 1 examples of Euler's transformation, ( l} n ( D n that of - - '- and s - + > were anticipated in 144. For further appli- W -J- 1 Cl Wr j 1 cations it is essential to know which null sequences are fully monotone. We may prove, in this connection, by repeated application of the first mean value theorem of the differential calculus ( 19, Theorem 8), the following theorem: Theorem 2. A (positive) sequence a Q , a^ ... is fully monotone decreasing if a function f (x) exists, defined for x ^ 0, and possessing differential coefficients of all orders for a;]>0, for which f(n)=a n while the kth derived function has the constant sign (- 1)*, (k = 0, 1, 2, . . .). Accordingly the numbers for instance, form fully monotone decreasing sequences; and from these many further sequences of this kind may be deduced, by means of the Theorem 3. // the numbers a , a lt ... and b , b l , . . . constitute fully mono- tone decreasing sequences, the same is true of the products a Q b Q , a L b L , a^ b , . . . Proof. The following formula holds, and is easily verified by induction relatively to the index k: It shows that, as required, all the differences of (a n b n ) are positive, if those of () and (b n ) are so. The following may be sketched as a particular numerical example: The series g 35. Applications of the transformation of series to numerical evaluations. 265 has extraordinarily slow convergence; in fact, it converges with practically as small a rapidity as Abel's series 2 1 In (logn) 2 . Yet by means of Eulers transformation, its sum may be calculated with relative ease. If we use only the first seven terms (to - --- inclusive), we can deduce the first seven terms of the trans- \ log lo / formed series. If we use logarithms to seven places of decimals, we find, with 6 decimals secured, the value 221 840 . . . for the sum of the series 88 . C. Markoff's transformation. As the choice of the array (A), p. 241, from which Markoffs transformation was deduced, is largely arbitrary, it is not surprising that we should be unable to formulate general theorems as to the effect of the transformation on the rapidity of the convergence. We shall therefore have to be content with laying down somewhat wider directing lines for its effective use, and with illustrating this by a few examples : Denoting as before by 2z^ the given series (assumed convergent), we choose the terms of the th column in our array (A) to be as near as possible to those of the given series, and at the same time to possess a sum s (0 ^ which we can indicate by a convenient closed expression; this is analogous to the condition of Rummer's trans- formation. The series 2(z^ a o^) now certam ty converges more rapidly than 2z (1c) ; proceed with this new series in the same way, for the choice of the next column in our array, and so on. The effect of the transformation will be similar to that of an indefinitely repeated Rummer's transformation, the possibility of which was already indi- cated in the examples 153, 2 a (cf. Ex. 130). 00 I As an example, we may take the series -~ , which is practically useless 156. &=i* for the direct evaluation of its sum -. Here we think of the th row and b column as consisting entirely of noughts, which we do not write down The choice of the series 5] -- for the first column, which was already used K (K -f- 1) on p. 247, then appears obvious enough. This gives 2? <.-.,*) -2* As second column, we shall then, as in 153, 1, choose the series and so on. The k ih row of the array thus takes the form 1 _ 0! _ ___ ![_ _ __ 2! __ /e 3 " k (k+ 1) + k (k + 1) (k + 2) + k (k + 1) (k f- 2) (k + 3) + ' ' ' (k hxcd) ' The further calculations are, however, simplified by breaking off this series at the (k l) th term and adding as # h term the missing remainder r k , after 38 This example is taken from the work of A. A. Markoff: "Differenzen- rechnung", Leipzig, p. 184. 1890. 266 Chapter VTIT. Closed and numerical expressions loi the sums of series. which the series is regarded as consisting entirely of noughts. The /5 th row now has the form: * 0' .... ,__(^)' ^ ~ Subtracting the terms of the right hand side from the left in succession, we easily find ___ ... (2 /*-!)* In our case, the process of splitting up the scries ^ into an army of the form (A) of p. 241 thus gives: 1 = 1 1 0! , I" 2 J 2 3 1 2* 3 1 0! 1! 2! " 3 4 8-4.* > ! 3 3 -4-5 1 __._ 01 _ + 1! I . . . _L I (ft-1)! ft 8 k (k + 1) ' I fc (* 4- 1) (ft + 2) ' ' ft (ft + 1)... (2 ft -1) r ft a (A :+l)...(2ft-l) Since all the terms of this array arc g> 0, the main rearrangement theorem OO itself shows that we may sum in columns and must obtain - as ultimate result. Now in the n tb column we have the series ' (nfixed) - By 132,3 for a = n+l and /?--=M, the series in the square brackets has the sum 1 n (n -f 1) . . . 2~n ' Hence the n ih column has for sum s (n) ... ( n _ J\ J _ * ____ j __ * __ 1 k ; Ln a (n+l)...(2n- l)^n (n+ 1) . . . 2 wJ = 3- 1 Therefore we have This formula is significant not only for numerical purposes, in view oi the appreciable increase in the rapidity of the convergence, but almost more so because it provides a new means of obtaining the closed expression for the sum of the series , which we only succeeded in determining indirectly K by using the expansion in partial fractions as well as the series expansion of the function cot. In fact we can easily establish directly (cf. Ex. 123), that 123 implies the expansion, for | x \ < 1 : Exercises on Chapter ViTl. 267 Putting x -~ , we at once deduce 39 Li y-L = 3 v(!Li A further application of fundamental importance of Markoffs transforma- tion we have already come across (v. 144) in Euler's transformation, which was indeed deduced from Markoffs. For further applications of Markoffs transformation we must refer to the accounts of Markoff himself (v. p. 265, footnote 38) and of E. Fabry (The'orie des series & termes constants, Paris 1910). Their success depends for the most part on special artifices, but they are sometimes surprisingly effective. Nu- merous examples will be found, completely worked out, in the writings re- ferred to. Exercises on Chapter VIII. I. Direct formation of the sequence of partial sums. for \x\<l t 100. a ) + + + + ....- for .- for ) is ' for a " positive " g-ent When does the series still continue to converge for arbitrary a n , and what is its sum? 1058. a) J tan- 1 ^ = ; n=i n * (hint : tan~ l - - tan~ * - - = tan~ x - ] . n I w-f-1 n"J , ~ t ~ _ " 6 (6 + ij ~~ 6 (b + 1) (6 + 2) 6 - 1 ' every Ay >>0and^- is divergent. Cf. a note by /. Schur and the author: r 'Cber die Herleitung der Glei- J? e p 174. 1918. chung J? e=-^- M , Archiv der Mathematik und Physik, Sen 3, Vol. 27, 268 Chapter VIII. Closed and numerical expressions for the sums of scries. 105- ) ?"; (hint : cot y tan y = 2 cot 2 y) . ' let ft- ft- - ft denote fixed given natural numbers, all different, and a =)= 0, 1, 2, ... any real number, while g (x) denotes an integral rational function (polynomial) of degree <& 2. We assume the expansion in partial fractions: The given series then has the sum -tr! v La a-f- 1 * * " + *> ! J 1^ 1 __1 = JL/ H9\ 107. a ) i. 2 .6~7 ~3-4 8."9 + 5.6 10-11 H ~ 60 \* 60 >/ ' _ _ ___ _ ___i I~2 4-5 ~ 3 4.6 7 + 5. 6- 8- 9 H 36 6 I 1 I = 1 G.7^5.6-8-9^" 36' . 5 a +~~ + '* 5 4""l 4 + 1) ~~ 2 r(4T2MlT) + 3^14"^" fT) ~ H = J ff 2 "" 2" J h > ir^ + + T + '-'-S^-T 10 ' 8 - II. Determination of closed expressions by means of the expansions of elementary functions. 108. a) l_^_-^ i + n l p + _l . . 1 1 , 1-8 1 , 1-3-5 1 , b > ' + ' + + -" n - ; gives for y = 7 and 2=1,2,3: -. i i i i 7 Exercises on Chapter VJI1. 269 c) _1 L_. + __ 1 .__+... ^JL^-S); 1 1 1 JT a I J\ XT V ' ' ^^-j~i"~~"o" - - - 1 ' 2 ' n =l( 4n3 -"^ a 16 32-3g __ __ j. __ -a.- 5 + 7 II" 1 " 13 17 + 1 1.2 1-2 3 b) " _._ ....^. 2-3 4^3.4.5.6^4.5.6.7.8^ 18 2' e) 111. If we write jj f- ^ ! , ,V= ntlo V (^ + w ) -V , then n * ^ 2 39 5 ,197 . If we \\rite ^ - = g p e, (p = 1, 2, . . .), then the numbers j p are n=l MI iniegers obtainable by the symbolical formula g p+1 = (1 + g) p . We have g a = 1 = 2, g'g = 5, . .. . 113. x may be summed in the form of a closed expression by means of elementary functions when x\y is a rational number. Special cases are: 1 270 Chapter VIII. Closed and numerical expressions lor the sums of series. 114. Writing- J T -^-- = L (x) , (|a:|<l), we have, if (x n ) denotes n=i l ~~ x Fibonacci's sequence 6, 7, oo 1 oo / _ 1)* 1 ft s _ And if we write y, - = s, and \7 - - - = 5 ', we have f = ^ 5 . A=ia 2 jb-i ^-! *~* 5 III. Exercises on Euler's transformation. 115. We have (for what values of #?) (-i" , * = (-1)" n=0 v* 116. If we put ,x e n=0 we have b n A" a Q . In particular, therefore, n=l "*"' n=L 117. Quite special cases are: 118. If A n a = b n , then J n 6 = a n . What accordingly are the inverse equations to those of the preceding- exercise? 119. If (a n ) be a null sequence with (p -f- l)-fold decreasing monotony (p ]> 1), oc the sum s of the series 2 ( l) n a n satisfies the inequalities n A a, ~ >. 2 -r 2 ^ ^ ^ 2 P 2 ^ 2 s ^ T 2 P 2^ Use this to prove the equality n x hm Exercises on Chapter VIII. 271 12O. If s/c and S n denote the paitial sums of both the series in 144, we have Use this relation to prove the validity of Euler's transformation. 121. The following relations hold, if the summation on cither side is taken to start with the index and the difference-symbols A operate on the coefficients on the left, a/ ,&, _>fc + t respectively: a-y n Wllh b) 27(-l)* afc * 8 *=(l-y 9 )274 ll a .y ait with c) 27 (- 1) ***+! z 2 * + ' = V i -y a 27 A " <vy an f * witn 122. Thus e. g^. 2 a: 9 2- - 3" iTS'+s Putting- a; = - , -, , -=- , j- , _ Q , ..., this provides peculiarly convenient O f J- 1 f <y series for jt, as for instance == 2 tan- 1 --- f- tan" 1 - = 5 tan" 1 - -J-Stan- 1 ^, 4 3 i i t <j and others. . The preceding series for tan- 1 x may also be put m the form 2 2-4 _ y-' Hence deduce the expansion IV. Other transformations of series. oo 1 124. Writing 1 V --- S pj we have a) S a +5 a + 5 4 + -..==l ; b) 5 2 4 - a ... = ... = l; d) S 4 + ~S 4 + ~5 6 + ... 4--.- = ^; t) S J -^S 4 +Is 6 - + --- h) "2"^"""3' 58 + ---- where C denotes Euler's constant, defined in 128,2 and Ex. 85 a. 272 Chapter VIII. Closed and numerical expressions for the sums of series. 125. With the same meaning for S p as in the preceding* exercise, writing /I 1\^ . ,. n { T - r TTT ) = b k and hm \*+l 2k) * n we have &a S 2 + b a 5 J H ---- = 1 log A. (The existence of the limit A results from the convergence of the series. We have A = \/ %n .) 126. , ~ _ i _ = r i_ _ j_ i , iji __ *' + - " l 11 128. With reference to 35 A, establish the relation between ~ __ 1 _ ~ 1 ^ " < 3 n <* - - n * > - I 3 an - _ __ __ - - ( n + * + /> - I) 3 n -Si (n -f a - I) 3 . . . (n -f o: + p) and, by giving- special values to a and p, prove the following transformations: V JL _ A 4. 25 _ ^ " '' 1 ' = 8 _ 9 133 3 4_ 4 ~ ___ 1 __ ~ 8 + 2.3 3+ "5 ^ n* (n -f I) 3 (n + 2) (n -f- 3) 3 (n + 4) ' -^ f, 1 _83 __ 2^3* ~ ___ 1 ____ n=l W ( n + X ) 8 " 63< ^ 35 n-l n3 ( w + i ) 3 ( M + 2 ) 3 ( w + 3 ) 8 " Evaluate the sum of the first series to 6 places of decimals. 129. Prove similarly the transformations: _ 6 6 (+!) ' .. v-J _!._ '~ tl^r* i a (- i)" 4 n f, Evaluate the sums of the series a) and b) to 6 places of decimals. Exercises on Chapter VIII. 273 130. a) Denoting by T P the sum of the series c) in the preceding exercise, we obtain relations between T l and T. 2 j c + l9 T z and T^+y. What are these relations? Is the process &->QQ allowed in them? What is the transformation thus obtained? Is it possible to deduce it directly as a Marhoff transformation? b) We have - - ( - 1) "~ t - 1 1 v (-!)"-'_ a 1 -- Give the form now taken by the transformations of the series for log 2 which were indicated in a). c) Carry out the same process with the series 122 for . 131. The sum of the series J5~ ; - ; - 7* - rs where n starts from ^ n logn log- 2 n ( Io 8s n ) the first integer satisfying log a n > 1, evaluated to 8 decimal places, is exactly ^ 1-00000000. How may we determine whether the actual decimal expan- sion begins with . . . or with 1 . . .? The solution of this problem requires a knowledge of the numerical value of e"' = e (e ) to one decimal place at least; this is "' = 3814279-1 ... It suffices, however, to know that e'" - ["'] - 0-1 ---- (Cf. remarks on p. 249.) 132. Arrange in order of magnitude all natural numbers of the form p q , (p, q positive integers ^> 2) and denote the nth of the numbers so arranged by p n > s tl iat (p lt p t , ...) = (4, 8, 9, 16, 25, 27, 32, . . .). We then have v- (Cf. 68, 5.) Part III. Development of the theory. Chapter IX. Series of positive terms. 36. Detailed study of the two comparison tests. In the preceding chapters we contented ourselves with setting forth the fundamental facts of the theory of infinite series. Henceforth we shall aim somewhat further, and endeavour to penetrate deeper into the theory and proceed to give more extensive applications. For this purpose we first resume the considerations stated from a quite elementary standpoint in Chapters III and IV. We begin by examin- ing in greater detail the two comparison tests of the first and second kinds (7 and 73), which were deduced immediately from the first main criterion (70), for the convergence or divergence of series of positive terms. These, and all related criteria, will in the sequel be expressed more concisely by using the notation 2c n and 2 d n to de- note any series of positive terms known a priori to be convergent and divergent respectively, whereas 2a n shall denote a scries alsoy in the present chapter, of positive terms only whose con- vergence or divergence is being examined. The criterion 73 can then be written in the simple form 157. (Ij a n <:c n : ' <B, a n ^d n : <3>. This indicates that, if the terms of the series under consideration satisfy the first inequality from and after a certain n, then the series will converge] If, on the other hand, they satisfy the second inequality, from and after a certain n, then it must diverge. The criterion 73 becomes in the same abbreviated notation 158. (H) ^ : , n Before proceeding we may make a few remarks in this connexion. But let us first insist once more on one point: Neither these nor any 274 36. Detailed study of the two comparison tests. 275 of the analogous criteria to be established below will necessarily solve the question of convergence or divergence of any particular given series. They represent sufficient conditions only and may therefore very well fail in special cases. Their success will depend on the choice of the comparison series 2c n and 2 ' d n (see below). The fol- lowing pages will accordingly be devoted to establishing tests, as numerous and as efficacious as possible, so as to increase the pro- bability of actually solving the problem in given special cases. Remarks on the first comparison test (157). 159. 1. Since for every positive number g the series 2gc n and 2gd n necessarily converge and diverge respectively with 2c n and 2 d n , the first of our criteria may also be expressed in the form: 5^*(<+oo) : , >g(>0) : S> or, even more forcibly, in the form hm ?- < -f oo : S, lim^>0 : S>. c n a n 2. Accordingly we must always have: hm == + oo , lim - "- = c n or, otherwise expressed: lim- a *== -[-oo is a necessary condition for the divergence of ~fl w , lim " = necessary n n w convergence 2& n * 3. Here, as in all that follows, it is not necessary that actual unique limits should exist. This may be inferred, to take the question quite generally, from the fact that the convergence or divergence of a series of positive terms remains unaltered when the series is sub- jected to an arbitrary rearrangement (v. 88). The latter can in every case be so chosen that the above limits do not exist. For instance 2c n can be taken to be 1 + + 5 +" J ~K* * "> anc * ~ a n to ^ e ^ le se " cs 2 + l + 8+4~l~82~l~l6H ' obtained from the former by interchanging the terms in each successive pair; the ratio certainly tends to no unique limit; in fact, it has c n distinct upper and lower limits 2 and |. Similarly, let 2 d n be chosen to be the series 1 + | + | + | + ***, and let 2a n be the series 276 Chapter IX. Series of positive terms. deduced from the former by rearrangement, (in this series every two odd denominators are followed by one even one). Here ~ has the two distinct upper and lower limits | and |. In a similar manner we may convince ourselves by examples in the other cases that an actual unique limit need not exist. //, however, such an unique limit does exist, it necessarily satisfies the conditions indicated for lim and lim, since it is then equal to both. 4. In particular: No condition of the form -^ is necessary for the convergence of 2a n unless all the terms of the divergent series 2 d n remain greater than a fixed positive d. For, even if we only have lim d n = 0, by choosing so that and writing a h = d k , a n = or = the corresponding term c n of any convergent series c n for every other n, we evidently obtain a convergent series a n , but it is equally evident that -~ does not >0. 160. Remarks on the second comparison test (158). 1. The validity of the comparison test II may now be established more concisely as follows: In the case marked (G), we have, from and after a definite n, _!L^>.JL f i. e. (--) is a monotone descending sequence, whose limit c n c + i \ c nJ y is defined and ^ 0. In particular lim = y < + oo, and, by 159, 1, 2a n is convergent In the case marked (3)), K* J is monotone ascend- ing from and after a particular n, and accordingly also tends to a definite limit > 0, or to -\-oo. In either case the condition lim -, n - > n of 159, 1 is fulfilled and this shows that 2a n is divergent 2. The comparison test II thus appears as an almost immediate corollary to the comparison test I. If the convergence or divergence of a series 2 a n can be inferred by comparison with a (definitely chosen) series 2c n or Sd n in accordance with 158, then this may also be inferred by means of 157 (or 159, 1), but not conversely, i. e. if I is decisive, II need not be so. Examples of this have already occurred in the pairs of series of 159, 3. For the first pair we have lim = 2, while ^ alter- 36. Detailed study of the two comparison tests. 277 natelv ~ 2 and -= --, i. e. it is sometimes greater, sometimes less than the o corresponding ratio JL * Li , since this constantly = ~. The second C n 6 pair of series represents an equally simple case. 3. This relation between the two types of comparison tests be- comes particularly interesting when we come to deal with the two tests to which we were led in 13 as immediate applications of the first and second comparison tests. These \\ere the root and ratio tests, inferred from I and II by the use of the geometric scries as com parison series, and they may be stated thus: ,-(^*<l : <S : 3) Our remark 2. shows that the ratio test may very well fail when the root test applies (the series 2 a n given there are obvious examples of this). On the other hand our remark 1. shows that the root test must necessarily work, if the ratio test does so. This relation between the two comparison tests is expressed in more significant form by the following theorem, which may be regarded as an extension of 43, 3. Theorem. // x 19 x^ 9 ... are arbitrary positive terms, we always 161. have l H. iVaL; Proof. The inner inequality is obvious 2 , and the two outer in- equalities arc so closely similar that we may be content with proving one of them. Let us choose the right hand inequality and put so that the statement reduces to "/* <//". Now if // = + oo, there is nothing to prove. But, if // < -f- oo, we may, given e > 0, assign an integer p, such that, for every v ^> p, we always have 1 This theorem is of the same character as 43, 3. In fact, writing 1 ^i y&> * or tne rat i s ~ ~ ' we are concerne ^ w *h a com- parison of the upper and lower limits of y n and of y n f = y y t y a . . . y n . fl For this reason, it is usual to write more shortly: ' implying that in the centre, either lim or lim may be considered indifferently. Such an abbreviated notation will frequently be used by us in the sequel. 10 (G51) 278 Chapter IX. Series of positive terms. This inequality may be supposed written down for every v ~ f> , p -f- 1, . .., n 1, and we then multiply all these inequalities together, deducing, for n > p, Let us, for brevity, denote the constant number x * (p,' -f- ~ J by A\ then, for n > />, we always have But V0f-*l, and hence (// + -0 %A -> p' + ~ . We can therefore so choose n >p that, for every n>n n , we have f// + ~J \ A < /i' + 6. We then have a fortiori, for every n > n , and hence also /i < // + e, or, as asserted, since e is arbitrary, /i <[ /e r . (Cf. p. 68, footnote 10.) Moreover we can show by simple examples that the sign of equality need not hold in any of the three inequalities of 161, which is now completely established. 4. The preceding theorem shows in particular that, if lim ~ n exists, lim yx n must also exist and have the same value. Hence in particular: If the ratio test works in the form given in 76, 2, then so will the root test, necessarily, (but not the converse!). To sum up: The ratio test is theoretically less powerful than the root test. (Never- theless it may frequently be preferred, as being easier of application.) 5. In this place we have also to refer to the remarks 75, 1 and 76, 3. 37. The logarithmic scales. We have already observed that such criteria as those just dis- cussed only provide sufficient conditions and may accordingly fail Li particular cases. Their efficiency will depend on the nature of the chosen comparison series 2c n and 2d n ; in general terms we may say that a G-test will present a better prospect of success the greater the magnitude of the c n 's, a 3) -test, on the contrary, the smaller the magnitude of the ^ n 's. In order to express these circumstances more precisely, we proceed first to define the concept of the rapidity of convergence: A convergent series will be said to converge more or less rapidly according as its partial sums approach more or less ra- pidly to the sum of the series; and a divergent series will be said to diverge more or less rapidly in proportion to the rapidity with which its partial sums increase. More precisely: 37. The logarithmic scales. 279 Definition 1. Given two convergent series 2c n = s and c n ' = s ' of positive terms, whose partial sums are denoted by s n and s n ', the corresponding remainders by s s n = r n , s's n ' = r n ', we say that the second converges more or less rapidly (or better or less well) than the first, according as lim^ = or lim-?'=+oo. r n r n If the limit of this ratio exists and has a finite positive value, or if it be known merely that its lower limit > and its upper limit < -f-cx), then the convergence of the two series will be said to be of the same kind. In any other case a comparison of the rapidity of convergence of the two series is impracticable 3 . Definition 2. // 2 d n and 2d n ' are two divergent series of posi- tive terms, whose partial sums are denoted by s n and s n ' respectively, the second is said to diverge more or less rapidly (or more or less markedly) than the first according as s ' s ' lira = + oo Of lim - - = 0. , ^ *n If the upper and lower limits of this ratio are finite and positive, then the divergence of both series will be said to be of the same kind. In any other case we shall not compare the two series in respect of rapidity of divergence 4 . The two following theorems show that the rapidity of the con- >ergence or divergence of two series may frequently be recognised from the terms themselves (without reference to partial sums or re- mainders): c ' Theorem 1. // -^- *0(+ oo), then 2c n ' converges more (less) rapidly than 2c n . Y / r / 8 In the case lim - - = (> 0) and lim < -f oo (= + OO), we might also speak of the series 2c n ' as "no less" ("no more") rapidly convergent than the series S c n \ this however presents no particular advantages In the case of the lower limit being and the upper limit -}-OO, the rapidity of the conver- gence of the two series is totally incommensurable. A similar remark holds for divergence. (The student should illustrate by examples the fact that all the cases mentioned can really occur.) These definitions may be directly transferred to the case of series of arbitrary terms, replacing r n and r n ' by their absolute values. 4 The properties referred to in these definitions are obviously transitive^ i. e. if a first given scries converges more rapidly than a second, and this again more rapidly than a third, the first series will also converge more ra- pidly than the third. 280 Chapter IX. Series of positive terms. Proof. In the first case, given e, we choose n so that for every n > n n we have cj < . We then also have \t n n ',i c + i + c Consequently this ratio tends to 0. The second case reduces to the first by interchanging the two series (cf. the theorem of 40, 4, Rem. 4). This proves all that was required. Theorem 2. // -,- *0(-{~oo), then 2d n ' diverges less (more) rapidly than 2 d n . Proof. By 44,4 it follows immediately from -j~ *0 that This proves the statement. 163. Simple examples. 1. The series 1 v. 1 ^ 1 v. 1 ^ I _ 3 -- ^J M ,> </ n * y? J^ yi _ x _ yr * yr x V N T __ V , ^/ 2 ^ ^ , ^j 9n , ^ qli ^^ i -Z/ are such that each converges more rapidly than the preceding 1 . In fact we have e. g. for n > 3 : JL ^_ L_^ s-ss 9 n\ ' 3""! which tends to 0. Similarly -- -> (by 38,4); the other cases are even n simpler. 2. The series 1 7 ~ - are such that each diverges less rapidly than the preceding. Besides the above simple examples, the most important cases of series with rapidity of convergence forming a graduated scale are afforded by the series which we came across in 14. As we saw in that paragraph, the series y ~~ / M) a converge for a > 1 and diverge for a < 1 . Our theorems 1 and 2 now show more precisely that when p is fixed each of these series will converge or diverge less and less rapidly as the exponent a approaches unity (remaining > 1 in the first case and <^ 1 in the second). Simi- larly each of these series will converge or diverge less and less ra- 37. The logarithmic scales. 281 pidly, as p increases, whatever positive 5 value may be given to the ex- ponent a (> 1 in the first case, < 1 in the second). The second alone of these statements perhaps requires some justi- fication. Divide the generic term of the (p + l) th series with the ex- ponent a' by the corresponding term of the /> th series taken with the exponent a. We obtain In the case of divergent series, a, and ef are positive and ^1; the ratio therefore tends to 0, q e. d. In the case of convergence, i. e. a and a! both > 1, the ratio tends to -f- oo; in fact, by reasoning analogous to that of 38, 4, we have the auxiliary theorem that the numbers (loRp Kn ^ {log(lo gj> n)} flf (\og p nf (log p nf form a null sequence, /? = a 1 denoting any positive exponent and p any positive integer. This proves all that was required. The gradation m the rapidity of the convergence and divergence of these series enables us to deduce complete scales of convergence and divergence tests by introducing these series as comparison series in the tests I and II (p. 274). We first immediately obtain the fol- lowing form of the criteria: (1) "= -^ with ^ ' ~ ^-w I * lror -ML lrnr wi (\r\cr~\ ii\ ' I vv J^^ A fi> ~ 164. logn ., with These criteria will be referred to briefly as the logarithmic tests of the first and second kinds also in the case p = Q. Their effi- ciency may be increased by the choice of p, and, for fixed p, by the choice of a, in accordance with our previous remarks 6 . * For a = /?<0, each series of course diverges more rapidly than the preceding one with the exponent replaced by 1; thus e. g. ^ & ~ , with /?>>0, diverges more rapidly than JSJ' e The convergence and divergence of series of the above type was known to N. H. Abel in 1827, but was not published by him (CEuvres II, p. 200). A. de Morgan (The differential and integral calculus, London 1842) was the first 282 Chapter IX. Series of positive terms. For practical purposes it is advantageous to give other forms of these criteria. Such transformations are given below with a few remarks appended, but without completely carrying out the necessary calculations. Transformation of the logarithmic tests of the 1 st kind. 1. When a and b are positive, the two inequalities a ^ b and log a 5C log & are equivalent; after a slight alteration the inequalities 164, I accordingly become: logan -f log** + Iog 2 n !- + logyn i <; - p < : <o ( } log^ \^0 : S5. 2. Denoting for a moment by A n the expression on the left of (I'), we have, in (I") hm A n < : <2 , Km ^ n > : >, a test of practically the same effect. The parts relating to convergence are indeed completely equivalent in (I') and (I"); that relating to divergence is not quite so powerful in (I") as in (I'), since it is required in (I") that A n should remain, from some value of n onwards, not merely = but greater than a fixed positive number 7 . 3. If we use the somewhat more explicit notation A n A ( \ and consider both A ( * } and A** l) , we obviously have A (p fl) = 1 And since, by 38, 4, g ~ - = - -~^ - - tends as n increases to -f oc , this simple '\og v+1 n log(logj,w) transformation leads to the following result: If for a particular p one of the limits of A n A ( * } is different from zero, it is necessarily oo for the following p, in fact + oc or GO according as the preceding p was positive or negative. More pre- cisely, if we denote by fi p and x v the upper and the lower limits of A n A ( \ for every p t then if we have, for any particular />, x p ^ /i p < 0, we have x^ -= n v+1 == oo, and if Pj> ^ * P > 0, we have //+, = x,, +1 == + oo. If, however, XP < 0, n p > 0, we have x p+l = oo, fi p+ i = -f oo. The scales of reference (I) thus lead to the solution of the question of conver- gence or divergence if, and only if, for a particular p, the values y. p and ^ have the same sign. If the sign is negative, the series converges; if positive, it to use these series for the construction of criteria. Essentially, these criteria are consequences of 164, I and II; numerous transformations of them were subse- quently published as special criteria, e. g. by/. Bertrand (]. de math, pures et appl., (1) Vol. 7, p. 35. 1842), O. Bonnet (ibid., (1) Vol. 8, p. 78. 1843), U. Dini (Giornale di matematiche, Vol. 6, p. 166. 1868). 7 It would clearly, however, be wrong to write the last 25-test in the form Hm A n ^ 0, since the lower limit may very well be without a single term being positive. 37. The logarithmic scales. 283 diverges ; if the two numbers have opposite signs for some value of p, then for all higher />'s we have and the scale therefore is not decisive. Similarly it fails when both numbers are zero for every p. Transformation of the logarithmic tests of the 2 nd kind. 1. The following Lemmas are easily proved: Lemma 1. For every integer p ^ 0,/or every real a and every sufficiently large , an equality of the form /log,, (/i_- 1)\ ^ 1 ____ a ____ * n \ log,, n ) ' w log w ... log,, n ri l holds, where ( n ) is bounded 8 . The index n is here assumed to start with a value from and after which all the denominators are defined and positive. We immediatelv infer that, for every integer p ^ 0, for every real a and every sufficiently large n, } l l^jrl^J- 11. (]2?f ^ "JlV* logn "* log^^n \ log p n j n wlogn n logti . . . log f ,_ 1 n log n . . . log^ti w 2 where (/) is again certainly bounded 9 . Lemma 2. Let 2 a n and 2 a n f be two series of positive terms; if the series whose n^ term is 4. , aJ ts absolutely convergent, the two given series are either both convergent or both divergent. In fact, we have yv> 1 for every v\ taking, then, any positive in- teger m, writing down the relations for j/ = m, m-fl, ..., w 1, and multiplying them together, we at once de- duce that the ratio *'/ for n > m lies between two fixed positive numbers. 8 An equality of the above form of course holds under any circumstances. In fact we can consider the numbers # n as defined precisely by the equation: n log n . . . log p n The emphasis lies on the statement that (<?) is bounded. The proof is ob- tained inductively, with the help of the two remarks that if (#') and (#") are defined, for every sufficiently large n, by (1 x n ) a 1 a x n $,/ a; n a and Ic they are necessarily bounded, provided (#) is a null sequence and the num- bers y n are in absolute value |> 1 , say. The interpretation in the case p is immediately obvious. 284 Chapter IX. Series of positive terms. The conditions of the Lemma are fulfilled, in particular, when the ratios -"-i* and -^-~ lie between fixed positive bounds and the series J converges. 2. In accordance with the above we may express the logarithmic test of the second kind e. g. in the following form 10 : n n log w nlog/t ... log p __itt n log n... '<! or, after a simple transformation, 169. |f!iI_i-|--L-| ---- -f._ - L- -- l-wlogw ...log.n L M n ' ' n log n . . . log nj e "1* < or, finally, denoting the expression on the left hand side for brevity by /? n , and slightly restricting the scope of the 3) -test (cf. 165,2), Tim'B w <0 : <5, }^_B n >Q : 3). Remarks analogous to those of 165, 2 hold here. 3. The developments of 165, 3 also remain valid, with quite unessential alterations. For, if we use the more explicit notation B n B^\ we have ob- viously And, as log ;j + 1 n 4-OO, we may reason with this relation in precisely the same manner as with its analogue in 165, 3. It is unnecessary to develop this in detail. 4. Still more generally, we may at once prove that a series of the form ^ e (a ~ 1)n . n a (log n) ai (Iog 3 n) a . . . (Iog 7 n) a * converges '/ and only if, the first of the exponents a, a , a,, . . . , a q which differs from 1 is > 1. The values of the subsequent exponents have no further influence. When the comparison scries is put into this form, Raabe's test ( 38) and Cauchy's ratio test appear naturally as the th and the ( l) th terms of the logarithmic scale. 38. Special comparison tests of the second kind. The logarithmic tests deduced in the preceding article are un- doubtedly of greater theoretical than practical interest. They afford in- deed a more profound insight into the systematic theory of the con- vergence of series of positive terms, but are of little use in actually testing the convergence of such series as occur in applications of the 10 Here we make the n th term of the investigated series 2'a n correspond to the (n l) th term of the comparison series, which, by 82, theorem 4, is allowable. 38. Special comparison tests of the second kind. 285 theory. (For this reason we have only sketched the considerations relating to them.) For practical purposes the first two or three terms, at most, of the logarithmic scales may be turned to account; from these we proceed to deduce by specialization a number of simpler tests, which were discovered at various times, rather by chance, and each proved in its own way, but which may now be arranged in closer connexion with one another. For p = the logarithmic scale provides a criterion already estab- lished by /. L. Raabe 11 . We deduce it from 169, first in the form : - ft < o : e or, as we may now write more advantageously, > 1 : s>. 17 - The tfery elementary nature and great practical utility of this cri- terion makes it worth while to give a direct proof of its validity: the (^-condition means that, for every sufficiently large n, where ft = a 1 > 0. Hence and therefore w n + 1 is the term of a monotone descending sequence, for a sufficiently large n. Since it is constantly positive, it tends to a limit y ^ 0. The series 2c n with c n = (n 1) a n n a n + 1 there- fore converges, by 181. Since a n <^ ~gC n , the convergence of S a n immediately follows. Similarly, if the -condition is fulfilled, we have - or n- Accordingly w# n + 1 is the term of a monotone increasing sequence and therefore remains greater than a fixed positive number y. As + !> , y > 0, the divergence follows immediately. If the expression on the left in 170 tends, when n *+(X>, to a limit I, it follows from the reasoning already repeatedly applied (v. 76, 2) that / < 1 involves the convergence of 2a n > and I > 1, its divergence, while / = 1 leads to no immediate conclusion. 11 Zeitschr. f. Phys. u. Math, von Baumgarten u. Kttinghausen, Vol. 10, p. 63, 1832. Cf. Duhamel, J. M. C.: Journ. de math, pures et appl., (1) Vol. 4, p. 214, 1839. 286 Chapter IX. Series of positive terms. Examples. 1. In 25 we examined the binomial series and were unable to decide there whether the series converged or not at the endpoints of the interval of convergence, that is, whether for given real as the series .4:0 - .!<-"(:) were, or were not, convergent. We are now able to decide this question. For the second series we have __ a ~ n + 1 ~ n-f-1 Since this ratio is positive from a certain stage on, it follows that the terms then maintain one sign; this we may assume to be the sign -}-, since changing the signs of all the terms does not, of course, affect the argument. Further, ac- cording to this, from which we at once deduce, by Raabe's test, that the second of our series converges for a>0, and diverges for -<0. For a = 0, the series reduces to its initial term 1. For the first series we have and, since this value becomes negative from some stage on, the terms of the seja^shave an alternating sign from that stage on. If now we supple a- " ^ " we^lhefore have >1 whenW we inllr'that ultimately the terms a n are non-decreasing. The series must therefore^diverge. If however we suppose a-f- 1 > 0, we have ultimately, say for every n^>m t 9 and the terms ultimately decrease in absolute value. By Leibniz's criterion for series with alternately positive and negative terms, our series must therefore converge, provided we can show that ( j-*0. If we write down the rela- tions (a) for m, m-\-\, ..., n l and multiply them all together, we deduce for every n ;> m l-ll|- II f 1 - Since, however, the product Jf (l " j , by 126, 2, 3, diverges to 0, a n must also ^0, and theref6re I J must converge. Summing up, we therefore have the following results relating to the binomial series: 00 /cc\ The series J ( ) x n converges if, and only if, either | * | <; 1 , or x = 1 n=o \ w ^ 38. Special comparison tests of the second kind. 287 and a > 12 , or x = +1 and a > 1. The sum of the series is then by Abel's theorem of limits always (1 -f #) a . // a is an integer and is non-negative \ then the series is finite and hence converges for each x. In all other cases the series is divergent. (An appreciable addition to this theorem is provided by 247.) 2. The following criterion docs not differ essentially from that of Raabe; it is due to O. Schlomilch : - 1 i In fact, in the case ($), we have, by 114, n+l ^ e > 1 , a n w from which the divergence follows by Raabe's test. In case (6) we have, ultimately, if a >a'>l. By 170, this involves convergence. If, in the logarithmic scale, we choose p = 1, we obtain a cri- terion of the second kind which, omitting the limiting case cc = 1, * we may write with 171. A direct proof of the validity of this criterion can be given as follows As in the proof of Raabes test we first put the criterion in the following form: // now the G- condition is fulfilled} since, as we may immediately verify by 114, a, * (n - l)log(n - 1) > - 1 + (n - l)logn, we have a fortiori Accordingly wlogna M + 1 is the term of a monotone descending se- quence and accordingly tends to a limit y^O. By 131, the series whose n ih term is must converge. As a n ^>-jC n , the same is true of 2a n . If, on the other hand, the 3)- condition is fulfilled, we have (n l)log(w !) nlogn-a n + a . For w>4-oo, however, the expression in square brackets -* (I* 12 For a = 0, see above Chapter IX. Series oi positive terms. (by 112, b), and is therefore negative for every sufficiently large n Hence for those w's the expression ttlogn*0 n+1 increases monotonel> and consequently remains greater than a certain positive number y As a ., ^- , y > 0, it follows that 2 a must diverge. n -r i - w Jog- w ' n Here again we may observe, as repeatedly in previous instances, that, if a n tends to a limit /, then / > 1 involves convergence, and / < 1 involves divergence, while, from / = 1, nothing can be directly inferred. Even this, the first properly logarithmic criterion of the scale, will rarely be actually applied in practice. In fact, the scries which are amenable to this test, and not already to a simpler one (Raabe's test, or the ratio test), occur exceedingly seldom; and as their convergence is no more rapid than that of 3J -- , (a>l), these series are M (lOg W) a useless for numerical calculation. It enables us, however, to deduce easily one or two other cri- teria. We will above all mention 172. Gauss's Test 18 : // the ratio ^^ can be expressed in the form where I > 1, and (?? n ) is bounded*-*, then 2 a n converges when a>l and diverges when cc < 1 . The proof is immediate: when a^l, Raabe's test itself proves the validity of the assertion. For cc = 1, we write a n n n log n \ and as now the factor in brackets tends to zero since (A 1) > 0, the series certainly diverges, by 171. Gauss expressed this criterion in somewhat more special form as follows: "// the ratio -~^ can be expressed in the form (k an integer ^ 1) then 2a n will converge when b 1 b^ < I and diverge when b^ &/ ^> 1." The proof is obvious from the preceding. 18 Werke, Vol. 3, p. 140. This criterion was established by Gauss in 1812. " Cf. footnote 8, p. 283. 38. Special comparison tests of the second kind. 289 Examples. 1. Gauss established this test in order to determine the convergence of the so called hyper geometric series . 1.2 - >(y+"l) 17278 = V ~ 1-2... n where a, /?, y are any real numbers l5 different from 0, 1, 2 ..... Here which shows in the first instance that the series converges (absolutely) for j#|<Clj an d diverges for|sc|>l. Accordingly it only remains to examine the values x 1 and x = 1 . This is analogous to the case of the binomial series, to which, of course, the present one reduces when we choose ft = y (= 1) and replace a and x by a and x. For x = 1 , we have n+l **+ This shows that for every sufficiently large w, the terms of the series have one and the same sign, which may be assumed positive. Gauss's test now shows that the series converges for + /ff y 1 < 1, i.e. for a + /?<y, but di- verges for cc -f- ft ^> y For a? = 1, the series has, from some stage on, alternately positive and negative terms, since *** + i -> 1 , i. e. is ultimately negative. The relation 18 with word for word the same reasoning as was employed in 170, 1 for the binomial series, now shows that the hypergeometric series will diverge when a + ft y > 1 converge when a + ft y < 1 . We have only to verify further that it also diverges when cc + ft y 1 , as this does not follow from precisely the same reasoning as before. If for every n > p >- 1 we have ~ 1 + with I ^n | < ^> ^r every n t then, assuming p chosen so large that p 2 Since on the right hand side we have the product of the first (n p) factors of a convergent infinite product of positive factors, it follows that |a n |, for all these values of n, remains greater than a certain positive number. The series can therefore only diverge. 15 For these values, the series would terminate or become meaningless. For w = 0, the general term of the series should be equated to 1. 10 As before, (#) denotes a bounded sequence of numbers. 290 Chapter IX. Series of positive terms. 2. Raabc's 6- test fails if the numbers <* n in the expression ~~r~~~ ~~~* though constantly > 1, have the value 1 for lower limit. In that case, writing = 1 -f fj nj the condition lim n /? n = -f CO is a necessary condition for the convergence of S a n . In fact, if n fi n were hounded, \ve should have " 7~ " ~~n~~n* and Sa n would be divergent by Gauss's test 17 - 39. Theorems of Abel, Dint and PrinffsJieim and their application to a fresh deduction of the logarithmic scale of comparison tests. Our previous manner of deducing the logarithmic tests invests these, the most general criteria yet obtained, with something of a for- tuitous character. In fact everything turned on the use, as comparison series, of Abel's series, which were obtained themselves only as chance applications of Cauchy's condensation test. This character of fortuitous- ness disappears to some extent if we approach the subject from a different direction, involving a greater degree of inevitablencss. Our starting point for this is the following 173. Theorem of Abel and Dini ls : If j d n is an arbitrary divergent n-l series of positive terms, and D n = d^ -f- d^ -J- -f- d n denotes its partial suniSy the series d n f converges when a > 1 _ n =i n=i D% diverges when a <^ 1 . Proof. In the case cc = 1, i I i ^ n " J h > As D,,-> + cx) by hypothesis, we can therefore choose k = ft n , for each n> so that 17 Cahen, E.: Nouv. Annales de Math., (3) Vol. 5, p. 535. " N. H. Abel (J. f. d. reine u. angew. Math , Vol. 3, p. 81. 1828) only proved the divergence of n * - J U. Dini (Sulle serie a termini positivi, An- nali Univ. Toscana Vol. 9. 1867) established the theorem in the above com- plete form. It was not till 1881 that writings of Abel were discovered (CEuvres 11, p. 197) which also contain the part relative to convergence of the theorem given above. 39. Theorems of Abel, Dini and Pringsheim. 291 by 81,2, the series 2 a n must accordingly diverge when = 1, and a fortiori when a<^l. The proof of its convergence in the case a > 1 is slightly more troublesome. We may at the same time prove the following extension, clue to Pnngsheim. Theorem of JPringshehn: The series 174. where d n and D n have the same meaning as before, converges for every Q > 0. Proof. Choose a natural number 6 such that <o. It then r P suffices to prove the convergence of the above series when the ex- ponent Q is replaced by T = . Since, further, the series 1 converges, by 131, since D n _ i ^ D n + -\-oo, and since its terms are all positive, it would also suffice to establish the inequality _. or !_: ^ D' that is to say, to prove that a-*') for every x such that < x < 1. But this is obvious at once, from (1 - x) = (1 - )(! + x + + a;"- 1 ). Therefore the theorem is established. Additions and Examples. 175. 1. In the theorem of Abel-Dmi, we may of course replace the quantities D n by any other quantities D n ' asymptotically equal to them, or for which the D ' ratio ~ lies between two fixed positive numbers, for every n (at least from some stage on). By 70,4 the convergence or divergence of the series 2 a* cannot be affected by this change. 2. By the theorem of Abel-Dim, v j t . v d n ~ " -- D. diverges with 2 d n . We may enquire what is the relation as to magnitude between the partial sums of the two series. Here we have the following elegant Math. Annalen, Vol. 35, p. 329. 1890. 292 Chapter IX. Series of positive terms. Theorem 180 . // ~~ - 0, we have ai '! d I A The new partial sums thus increase essentially like the logarithms of the old ones. y 1 d n Proof. It a; n =-^->0, we have, by 112, b, *!- \ The undefined number D we here assume = 1, also replacing the above ratio by 1 for all indices n for which a; n = 0. By the theorem of limits 44,4, since log >> + OO , we then have lg 7 " This proves the theorem. Further, it is at once clear that in the statement of this theorem, the numbers D n may on both sides be replaced by others D n f asymptotically equal to them. 3. These remarks now enable us to elaborate in the simplest manner the considerations indicated at the beginning of (his section: 00 a) The series d nt with d n = l, i. e. D n n, must be considered as the n=l simplest of all divergent series, for the natural numbers D n = n form the proto- type of divergence to 4-OO. The theorem of Abel-Dim then shows at once that the harmonic series converges for diverges for a <J 1 , and the theorem in 2. shows further that in the latter case we have for a = 1 , i-( C01 =i n a \ di\ (cf. 128,2). b) Now choosing for 2 d n , in the theorem of Abel-Dini, the series - newly recognised to be divergent by a), and replacing, as we may by Land 2., D n by > n ' = logw, we conclude that converges when a > 1 diverges when a < 1 . The theorem in 2. shows further that 1.1. .1 i ( C01 n (log n) a \ din 80 v. Cesaro, E.: Nouv. Annales de Math., (3) Vol. 9, p. 353. 1890. 21 This condition is certainly satisfied if the numbers d n remain bounded, hence in all the series which will occur in the sequel. 39. Theorems of Abel. Dini and Prmgsheim. 293 c) By repetition of this extremely simple method of inference, we obtain afresh, and quite independently of our previous results' Starting from, a suitably large index (e p -f- 1), the series 2a If converges when a > 1 , n log n . . . log p _ t n (log^ n) a \ diverges when a < 1 , whatever value is given to the positive integer p. The partial sums of the series for a = 1 satisfy the asymptotic relation fm , 4. A theorem analogous to 173, but starting from a convergent series, is the following: Theorem of Dfnl. If Sc n is a convergent series of positive terms, and r n _ l = c n -j- c n + 1 + - denotes its remainder after the (n l y th term, then v _n_ s v Cn converges when or < 1 , (c n -{- r n H t -| ---- ) a I diverges when a :> 1 . Proof. The divergent case is again quite ea v sily dealt with, since, for or = 1 , _?*__ 4. ... and for eveiy (fixed) n, this value may be made ^> - by a suitable choice Z of ^, as r^ -0. By 81,2 the series must therefore then diverge. For a> 1 this will a fortiori also be the case, since r n is <; 1 for every sufficiently large n. If, however, o: << 1, we may choose a positive integer p so that -< 1 -- , and it now suffices again because r n < 1 for n >> w x to establish the con- vergence of the series -l where T = . Now r n tends monotonely to and consequently 2 ( f> n-i~" r n) is cer " tainly convergent with positive terms. It therefore suffices to show that - that is to say (i- y *)Pd-y) But the latter relation is evident, since <C y < 1 . 22 If we wiite e =e', e e ' = e", . . ., c' (r) = c (|i+1) , ... and denote by [fi (v) J = ^ the largest integer contained in (<) ( ^ , we may say that the factors in the denominators of the terms of our series are all > 1, if n be taken to start from the value (^+1). aa v. footnote 18, p. 290. 294 Chapter IX. Series of positive terms. 40. Series of monotonely diminishing positive terms. Our previous investigations concerned for the most part series of quite arbitrary positive terms. The comparison series used for the con- struction of our criteria, however, were almost always of a much simpler nature; in particular, their terms decreased monotonely. It is clear that for such scries simpler laws altogether will become valid and perhaps also simpler tests of convergence may be constructed. We have already shown in 80 that if in a convergent series 2 c n the terms dimmish monotonely to zero, we have necessarily n c n ~> 0, a fact which need not occur in the case of other convergent series (even with positive terms only). Again, Cauchy's condensation test 77 belongs to the series we are considering. We propose to institute one or two further investigations of this kind and, in the first instance, to deduce for such series a few very simple and at the same time very far reaching criteria. Their con- vergence, as we shall see, is often very much more easily determined than that of more general types of series. CO 176. ! The integral test' 24 . Let 2 a n be a given series of monotonely n=l diminishing terms. If there exist a function f(x), positive and monotone decreasing for x ^> 1 , for which f(n) a n for every n, then 2a n converges if, and only if, the numbers are bounded* 7 *. Proof. Since, for (k 1)<^<^A, we have f(f)^a j( , and for *<*<* + !, f(t)a k , (A an integer 2* 2), it follows (by 19, Theorem 20), that fc+l k ! f(t)dt^a^ f f(t)dt (ft = 2, 3 f ...) * k-i Assuming these inequalities written down for A = 2, 3,..., n and added, we obtain n+i n f(t)dt , + 8 + + 94 Cauchy: Exerciccs mathem , Vol 2 p 221. Paris 1827. 85 By 70, 4 it is of course sufficient that f(n] should be asymptotically proportional to the terms a nl or that f(n} = cc n a n with a positive lower limit for . Instead of requiring- that / M should remain bounded, we can of op course also require that J /"(/) di should converge. The two conditions (by l 19, Def. 14) are exactly equivalent. 40. Series of monotonely diminishing positive terms. 295 From the right hand inequality it follows, as the integrals J n are bounded, that so are the partial sums of the series; from the left hand inequality the converse is inferred. This, by 7O, proves all that was required. Supplement. The differences (s n J n ) at the same time form a monotone decreasing sequence with limit between and a r In fact, we have fH- 1 whence the statement follows, since a a ^ s n J n ^> a^ J, I> 0. The limit in question is therefore certainly positive, if f(f) is strictly monotone decreasing. Examples and Illustrations. 1. This test not only enables us to determine the convergence of numerous series, but is also frequently a means of conveniently estimating the rapidity of their convergence or divergence. Thus e. g. we can see at once that for a > 1 the series n vi * must converge, whereas n 00 -I / j A , where / = = log n * + OO, n=l n J l i must diverge. But we learn further that, for a > 1 , nf k ( ii< n i < f J i < ^ +1 ^ < J 1^' n + l n and therefore T * ~ ~~_i "^ j For a = 2, this evaluation was already established on p. 260. In the same way the supplement to 176 gives a fresh proof of the fact that the difference is the term of a monotone descending sequence tending- to a positive limit between and 1. This was Euler's constant mentioned in 128, 2. Similarly, the supplement also shows that when < a < 1 , the difference i is the term of a monotone descending sequence with a positive limit less than 1. Therefore, in particular (cf. 44, 6), for <; a -< 1 : and it is easily seen that this relation holds equally when a < 0. 296 Chapter IX. Series of positive terms. 2. More generally, from logpi/( ifa=li we can immediately deduce, by the same method, the known conditions ot convergence and divergence of Abel's series. We have now three totally distinct methods of obtaining these. The supplement to 176 again affords us good evaluations of the remainders in the case of convergence, and of the partial sums in the case of divergence. 3. If f(x) be positive for every sufficiently large x, and possesses, for those x's, a differential coefficient equal to a monotone decreasing (also posi- tive) function with the limit at infinity, the ratio f'(x)lf(x) is also mono- tone decreasing. Since it follows that the integrals f tiw j~JW d '~ X f (/) dt and are either both bounded or both unbounded. Hence we conclude that the series 2f'(n) and J7-C'- J will either both converge or both diverge. In the case of divergence, when necessarily f(n) *-f-OO, we have f f (n\ V ' \ ' In fact, here n V ' \ ' convergent when a > 1 . [fool- /"(<) i L whence the validity of the statement can be directly inferred. These theorems are closely connected with the theorem of Abel-Dim. 2. A test of practically the same scope, and independent of the integral calculus in its wording, is Ermakoffs test 29 . 177. If f(x) is related to a given series 2 a n of positive, monotonely diminishing terms, in the manner described in the integral test, and also satisfies the conditions there laid down, then \ diverges l f(x) for every sufficiently large x. Proof. If we suppose the first of these inequalities satisfied for xx 9 we have for these x's 86 Bulletin des sciences mathe"m., (1) Vol. 2, p. 250. 1871. 40. Scries of monotonely diminishing positive terms. 297 Consequently t x x * (1 - 0) / f(t)dt *[ / f(t)dt - / f(t}dt] jX XQ e X ^^('S f(f)dt-S f(t)dt] Xo X e x n ^*/ f(f}dt. x, t X Thus the integral on the left, and hence also J f(f)dt, is, for every 3*o x>x , less than a certain fixed number. The series 2" n must there- fore converge, by the integral test. If, on the other hand, we assume the second inequality satisfied for x > afj, we have, for these a;'s, A comparison of the first and third integrals shows further that On the right hand side of this inequality, we have a fixed quantity y > 0, and the inequality expresses the fact that for every w(>ar 1 ) we can assign k n so that (with the same meaning for J n as in 176) By 46 and 5O, the numbers J n cannot be bounded and 2, a n therefore cannot converge * 7 . Remarks. 1. Ennahof/'s test bears a certain resemblance to Cauchy's condensation test It contains, in particular, like the latter, the complete logarithmic com- parison scale, to which we have thus a fourth mode of approach. In fact, the behaviour of the series nlogn ..\ is determined by that of the ratio a? It is not difficult to carry out the proof without introducing 1 integrals, but it makes it rather more clumsy. 298 Chapter IX. Series of positive terras. As this ratio tends to zero, when a > 1 , but *-f-OO, when a. < 1 , Ermako/fs test therefore provides the known conditions for convergence and divergenc** of these series, as asserted 28 . 2. We may of course make use of other functions instead of e x . If q> (x) is any monotone increasing- positive function, everywhere differentiate, for which <p (x) >> x always, the series 2a H will converge or diverge according as we have ( <. 1 5 for all sufficiently large x l s With Ermakoffs test and Cauchy's integral test, we have command over the most important tests for our present series. 41. General remarks on the theory of the convergence and divergence of series of positive terms. Practically the whole of the 19 th century was required to estab- lish the convergence tests set forth in the preceding sections and to elucidate their meaning. It was not till the end of that century, and in particular by Pringsheini's investigations, that the fundamental questions were brought to a satisfactory conclusion. By these researches, which covered an extiemely extensive field, a scries of questions were also solved, which were only timidly approached before his time, although now they appear to us so simple and transparent that it seems almost inconceivable that they should have ever presented any difficulty 20 , still more so, that they should have been answered m a com- pletely erroneous manner. How great a distance had to be traversed before this point could be reached is clear if we reflect that Eider never troubled himself at all about questions of convergence; when a series occurred, he would attribute to it, without any hesitation, the value of the expression which gave rise to the series 30 . Lagrange in 17 70 31 was still of the opinion that a series represents a definite value, provided only that its terms decrease to O 32 . To refute the latter 28 This also holds for = 0, if we interpret log^x to mean e*. 29 As a curiosity, we may mention that, as late as 1885 and 1889, several memoirs were published with the object of demonstrating- the existence of con- vergent series J c * for which -"-^" 1 did not tend to a limit! (Cf. 159, 3.) c n 80 Thus in all seriousness he deduced from - - = 1 4- x -f- x 2 -f- , that 1 x and l=l o Cf the first few paragraphs of 59. 81 V. CEuvres, Vol. 3, p. 61. ia In this, however, some traces of a sense for convergence may be seen, 41. General remarks on series ot positive terms. 299 assumption expressly by referring to the fact (at that time already well known) of the divergence of J? , appears to us at present superfluous, and many other presumptions and attempts at proof cur- rent in previous times are in the same case. Their interest is there- fore for the most part historical. A few of the questions raised, how- ever, whether answeied in the affirmative or negative, remain of sufficient interest for us to give a rapid account of them. A con- siderable proportion of these are indeed of a type to which anyone who occupies himself much with series is naturally led. The source of all the questions which we propose to discuss resides in the inadequacy of the ciiteria. Those which are necessary and sufficient for convergence (the main criterion 81) are of so general a nature, that in particular cases the convergence can only rarely be ascer- tained by their means. All our remaining tests (comparison tests or trans- formations of comparison tests) were sufficient criteria only, and they only enabled us to recognise as convergent series which converge at least as rapidly as the comparison series employed. The question at once arises : 1. Does a series exist which converges less rapidly than any other/ 178. This question is already answered, in the negative, by the theorem 175, 4. In fact, when Zc n converges, so docs 2c n ' = -r 2 -* though, V r n-l obviously, less rapidly than 2c n , as c n :c n ' = r_ l > 0. The question is answered almost more simply by /. Hadamard 2 *', who takes the series ~c n ' = ^(l/r n _ 1 V r n ). Since c n = r n _ 1 r w , the ratio c n '' = VV n - 1 + W n * 0. The accented series conver- ges less rapidly than the unaccented series. The next question is equally easy to solve: 2. Does a series exist which diverges less rapidly than any other? Here again, the theorem of Abel-Dini 173 shows us that when 2d n diverges, so does 2d n ' = 2j~> an d hence the answer has to be in the negative. In fact as d n : d n ' = D n * -|- oo, the theorem provides, for each given divergent series, another whose divergence is not so rapid. These circumstances, together with our preliminary remarks, show that 3. No comparison test can be effective with all series. Closely connected with this, we have the following question, raised* and also answered, by Abel**: 83 Actti mathematica, Vol. 18, p. 319. 1894. ). f. d. reine u. angew. Math., Vol. 3, p. 80. 1828 300 Chapter IX. Series of positive terms. 4. Can we find positive numbers p n , such that, simultaneously, a ) P~ & * I tj . . A 3 ., . . f convergence \ / r n n \ are su ff t cient conditions for \ T . } b) n a w ^ a > J ' 'I divergence J o/ Vy;y possible series of positive terms? It again follows from the theorem of Abel-Dini that this is not the case. In fact, if we put a n = --, a > 0, the series ^0 n necessar- r /I ily diverges, and hence so does a n '^~ > where s n = a L -}- + # n . But, for the latter, p n a n / =~^0. The object of the comparison tests was, to some extent, the con- struction of the widest possible conditions sufficient for the determination of the convergence or divergence of a series. Conversely, it might be required to construct the narrowest possible conditions necessary for the convergence or divergence of a series. The only information we have so far gathered on this subject is that a n > is necessary for convergence. It will at once occur to us to ask: 5. Must the terms a n of a convergent series tend to zero with any particular rapidity? It was shown by Pringsheim** that this is not the case. However slowly the numbers p n may tend to -{- oo, we can invariably construct conveigent series 2 c n for which KPn C n= + 00 ' Indeed every convergent series -Tc 7 /, by a suitable rearrangement, will produce a series 2 c n to support this statement 36 . Proof. We assume given the numbers p n , increasing to -|- oo, and the convergent scries 2c n '. Let us choose the indices x , n 2 , ..., n , . . . odd and such that and let us write c n = cj r -i, filling in the remaining c n 's with the terms c/, c 4 ' y ... in their original order. The series 2 c n is obviously a re- arrangement of c n '. But Pn C n>* whenever n becomes equal to one of the indices n 9 . Accordingly, as asserted, The underlying fact in this connection is simply that the behaviour or a sequence of the form (p n c n ) bears no essential relation to that of to Math. Annalen, Vol. 35, p 344. 1890 86 Cf. Theorem 82, 3, which takes into account a sort of decrease on the average of the terms a n , 41. General remarks on series of positive terms. 301 the series 2 c n i. e. with the sequence of partial sums of this series, since the latter, though not the former, may be funda- mentally altered by a rearrangement of its terms. 6. Similarly, no condition of the form lim p n d n > is necessary for the divergence of 2 d n , however rapidly the positive numbers p n may increase to -f-oo 37 . On the contrary, every divergent series 2d n ', provided its terms tend to 0, becomes, on being suitably rearranged, a series 2 d n (still divergent, of course) for which \imp n d n Q. The proof is easily deduced on the same lines as the preceding. The following question goes somewhat further: 7. Does a scale of comparison tests exist which is sufficient for all cases? More precisely: Given a number of convergent series v r (i) y r (2) y r (*) -^ n > ^ C n ' ' > ^ C n > ' ' each of which converges less rapidly than the preceding, with e. g. () > + ao, for fixed k. (The logarithmic scale affords an example of such series.) 7s it pos- sible to construct a series converging less rapidly than any of the given series? The answer is in the affirmative 38 . The actual construction of such a series is indeed not difficult. With a suitable choice of the indices n 19 w 3 , ..., w fe ,..., the series is itself of the kind required. We need only choose these indices so large that if we denote by r^ the remainder, after the n th term, of the series 2 c^ k \ for every n I> n l , we have r n ^ <[ -^ with c n (3) ^> 2 c n (l) 2 ^. i* M M y (3 ^ ^^ * M r (3) -^ o , -> Wt > W " r n < 02 W C tt -^ ^ ( - w^n fc >w fc . 1 n w C^'<~ n c^ A| >2c n w The series c n is certainly convergent, for each successive portion of it belonging to one of the series -2*c n (fc) is certainly less than the 87 Pringsheim, loc. cit. p. 357 88 For the logarithmic scale, this was shewn by P. du Bois-Reymond (J. f. J. reine u. ungew. Math., Vol. 76, p. 88. 1873). The above extended solution is due to /. Iladamard (Acta math., Vol. 18, p. 325. 1894). 302 Chapter IX. Series of positive terms. remainder of this series, starting with the same initial term, i. e < ^ (k = 2, 3, ...). On the other hand, for every fixed k, 2--* + OOJ in fact for n > n q (q > k) we have obviously ^ > 2* *. This proves all that was required. In particular, there are series converging more slowly than all the series of our logarithmic scale 39 . 8. We may show, quite as simply, that, given a number of di- vergent series 2d n (k \ & = 1, 2, ..., each diverging less rapidly than the preceding, with, specifically, ^ +1) -f-d n (fc) +0, say, there are always divergent series 2 d n diverging less rapidly than every one of the series 2 d^ '. All the above remarks bring us near to the question whether and to what extent the terms of convergent series are fundamentally dishn guishable from those of divergent series. In consequence of 7. and 8., we shall no longer be surprised at the observation of Stieltjesi 9. Denoting by (e l , e 3 , . . .) an arbitrary monotone descending se- quence with limit 0, a convergent series 2c n and a divergent series 2d n can always be specif ied, such that c n = e n d n . In fact, if e n *0 monotonely, p n = -- > -f- oo monotonely. The series ff whose partial sums are the numbers p n , is therefore divergent. By the theorem of Abel-Dini, the series == v P+I""P n=l ^ ii + i is also divergent But the series 2cz==2s d^~y]\- ) is "^""^ \ "Fn ** 4- 1 convergent by 131. The following remark is only a re-statement in other words of the above: 10. However slowly n *+oo, there is a convergent series 2c n and a divergent series 2 d n for which ^ n = p n c n . In this respect, the two remarks due to Pringsheim, given in 5. and 6., may be formulated even more forcibly as follows: 89 The missing- initial Urms of these series may be assumed to be each replaced by unity. 41. General remarks on series of positive terms. 303 1J. However rapidly 2c n may converge, there are always divergent series, indeed divergent series with monotonely diminishing terms of limit 0, for which Thus 2 d n must have an infinite number of terms essentially smaller than the corresponding terms of 2 c n . Conversely: However rapidly 2 d n may diverge, provided only d n *0, there are always convergent series 2c n for which lim^ = +00. We have only to prove the former statement. Here a scries 2 d n of the form V d =- -1- -4- 4- I- - -\~ - -4- 4- - ,1 ,1 , ,1 ,1 is of the required kind, if the increasing sequence of indices n I9 w 3 , ... be chosen suitably and the successive groups of equal terms contain respectively n 19 (n. n 1 ), (n 3 ~ w a ), ... terms. In fact, in order that this scries may diverge, it is sufficient to choose the number of terms in each group so large that their sum > 1, and in order that the se- quence of terms in the series be monotone, it is sufficient to choose n k > n k ^ l so large that r nfc < c n/ , _ j (ft = 1, 2, . . . ; n = 1) as is always possible, since c n *0. As the ratio has the value - for n = n k , it follows that km = 0, as required. n In the preceding remarks we have considered only convergence or divergence per se. It might be hoped that wiih narrower require- ments, e. g. that the terms of the series should diminish monotonely, a correspondingly greater amount of information could be obtained. Thus, as we have seen, for a convergent series 2c n whose terms diminish monotonely, we hiwe nc n *0. Can more than this be asserted? The answer is in the negative (cf. Rem. 5): 12. However slowly the positive numbers p n may increase to + oo, there are always convergent series of monotonely diminishing terms for which n Pn c n not only does not tend to 0, but has +00 for upper limit 40 . 40 Pnngsheim, loc. cit. In particular it was much discussed whether for convergent series of positive terms* diminishing- monotonrly, the expression nlogn>c n must -*(); the opinion was held by many, as late as I860, that n log n*c n * was necessary for convergence. 304 Chapter IX. Series of positive terms. The proof is again quite easy. Choose indices n < n. 2 < such that P*,>** (*=1,2,. and write 1 c = c a = ...== c ni = -^ , "' >M'YC i The groups of terms here indicated contribute successively less than - >* > to the sum of the series 2c , so that this series 2 2* 2" converge. On the other hand, for each n = w, we have so that, as was required, limn-p n .c n = +00. 13. These remarks may easily be multiplied and extended in all possible directions. They make it clear that it is quite useless to attempt to introduce anything of the nature of a boundary between convergent and divergent series, as was suggested by P. du Bois- Reymond. The notion involved is of course vague at the outset. But in whatever manner we may choose to render it precise, it will never correspond to the actual circumstances. We may illustrate this on the following lines, which obviously suggest themselves 41 . a) As long as the terms of the series -S"c n and 2 d n aie subjected to no restriction (excepting that of being > 0), the ratio ~ is capable of assuming all possible values, as besides the inevitable relation lim -^ = we may also have lim-^ = +00. - a n a n The polygonal graphs by which the two sequences (c n ) and (d n ) may be represented, in accordance with 7, 6, can therefore intersect at an in- definite number of points (which may grow more and more numerous, to an arbitrary extent). 41 A detailed and careful discussion of all the questions belonging to the sub- ject will be found in Pringsheim*s work mentioned on p. 2, and also in his writings in the Math. Ann. Vol. 35 and in the Munch. Ber. Vol. 26 (1890) and 27 (181)7), to which we have repeatedly referred. 42. Systematization of the general theory of convergence. 305 b) By our remark 11, this remains true when the two sequences (c n ) and (d n ) are both monotone, in which case the graphs above referred to are both monotone descending polygonal lines. It is therefore certainly not possible to draw a line stretching to the right, with the property that every sequence of type (c n ) has a graph, no part of which lies above the line in question, and every sequence of type (d n ) a graph, no part of which lies below this line, even if the two graphs are monotone and are considered only from some point situated at a sufficiently great distance to the right. 14. Notes 11 and 12 suggest the question whether the statements there made remain unaltered if the terms of the constructed scries 2 c n and 27 d n are not merely simply monotone as above, but fully monotone in the sense of p. 263. This question has been answered in the affirmative by H. Hahn^. 42. Systematization of the general theory of convergence. The element of chance inherent in the theory of convergence as developed so far gave rise to various attempts to systematize the criteria from more general points of view. The first extensive attempts of this kind were made by P. du Bois-Reymond 43 , but were by no means brought to a conclusion by him. A. Pringsheim** has been the first to accomplish this, in a manner satisfactory both from a theoretical and a practical stand- point. We propose to give a short account of the leading features of the developments due to him 45 . All the criteria set forth in these chapters have been comparison tests, and their common source is to be found in the two comparison tests of the first and second kinds, 157 and 158. The former, namely (I) =<? s e > a n ^d n : V, is undoubtedly the simplest and most natural test imaginable; not so that of the second kind, given originally in the form 42 //. Hahn, Dber Reihcn mit monoton abnehmenden Ghedern, Monatsheft f. Math. u. Physik, Vol. 33, pp. 121134, 1923. 43 J. f. d. reine u. angew. Math. Vol. 76, p. 61. 1873. 44 Math. Ann. Vol. 35, pp. 297394. 1890. 45 We have all the more reason for dispensing with details in this connexion, seeing Pringsheim's researches have been developed by the author himself in a very complete, detailed, and readily accessible form. 306 Chapter IX. Series of positive terms. In considering the ratio of two successive terms of a series we are already going beyond what is directly provided by the series itself. We might therefore in the first instance endeavour to construct further types of tests by means of other combinations of two or more terms of the series. This procedure has, however, not yielded any criterion of interest in the study of general types of series. If we restrict our consideration to the ratio of two terms, it is still possible to assign a number of other forms to the criterion of the second kind; e. g. the inequalities may be multiplied by the positive factors a n or c n without altering their significance. We shall return to this point later. Except for these relatively unimportant transformations, however, we must regard (I) and (II) as the fundamental forms of all criteria of convergence and divergence 46 . All conceivable special com- parison tests will be obtained by introducing in (1) and (II) all conceiv- able convergent and divergent series, and, if necessary, carrying ovit transformations of the kind just indicated. The task of systematizing the general theory of convergence will accordingly involve above all that of providing a general survey of all conceivable convergent and divergent series. This problem of course cannot be solved in a literal sense, since the behaviour of every series would be determined thereby. We can only endeavour to reduce it to factors in themselves easier to survey and therefore not appearing so urgently to require further treatment. Pringsheim shows and this is essentially the starting point of his investigations that a systematization of the general theory of convergence can be fully carried out when we assume as given the totality of all monotone sequences of (positive] numbers increasing to +00. Such a sequence will be denoted by (pj; thus < Po ^ Pi ^ P2 ^ and Pn -* + In principle, the problem is solved by the two following simple remarks: a) Every divergent series 2 d n is expressible in the form 2d n *zpo + (Pi JP ) H h (Pn Pn-l) H n=0 (each in one and only one way) in terms of a suitable sequence of type (p n ). Also, every series of this form is divergent. 46 Thus since (as seen in 16O, 1,2) (II) is a consequence of (I) - it is ultimately from (1) that all the rest follows. 42 Systematization of the general tneory of convergence. 307 b) Every convergent series 47 2c n is expressible in the form / * Mi/ 1 ! \i ./ * \ , V / c= I - - 1 + 1 - - I + + I - - I + * H ^ C ~\P Q Pj ^ \P { P,)^ ^\P n Pn+l)^ (each in one and only one way) in terms of a suitable sequence of type ( w ). Also, every series of this form is convergent. In fact, when these statements have been established, we have only to substitute, in the two comparison tests (I) and (II), respectively for c n and d n > to obtain in principle all conceivable tests of the first and second kinds: All particular criteria must necessarily follow by more or less obvious transformation from the tests so ob- tained; for this very reason, the former can never present anything fundamentally new. They become of considerable importance, how- ever, m that they give deeper insight into the connexion between the various criteria and state the latter in a coherent form, and also apply them in practice. Herein lies the chief value of the whole method. It would accordingly be well worth our while to describe the details of the construction of special criteria exactly; but for the reasons given, we shall abide by our plan of giving only a brief account. 1. The typical forms a) and b) must be regarded as undoubtedly 180. the simplest imaginable forms for convergent and divergent series. But we can obviously replace them by many other forms, thereby altering the outward form of the criteria in various ways. For instance, by the theorem of Abel-Dini 173, diverge with 2(p n /> n -i)> while at the same time, by Pringsheim's theorem 174, 2~ n -~^- and 2 -",, converge for Q > 0. With a few restrictions of little importance, all divergent and convergent series are also expressible in one of these new forms. 2. Since the only condition to be satisfied by the numbers p 9 in the typical forms of divergent and convergent series which we are 7 Unless the terms are all from some stage on. 48 The pi oofs of these two statements are so easy that we need not go into them further. 308 Chapter IX. Series of positive terms. considering, is that they are to increase monotonely to f- oo, we may of course write \ogp n , iog 2 /> n , ... or generally F (p n ) instead of /> n , where F (x) denotes any function defined for x > and increasing monotonely (in the strict sense) to +0 with x. This again leads to criteria which, though not essentially new, are formally so when the /> w 's arc specially chosen. It is easy to verify that the first named types of series diverge or converge more and more slowly, as />-> + oo more and more slowly; by replacing p n successively e. g. by logp n , Iog 2 /> n , . . . , we therefore obtain a means of constructing scales of criteria 4a . The case p n = n naturally calls for consideration on account of its peculiar simplicity; the development of the ideas indicated above for this particular case forms the main contents of 37 and 38. 3. A further advantage of this method is due to the fact that one and the same sequence (p n ) will serve to represent both a divergent and a con- vergent series. The criteria therefore naturally occur in pairs. E. g. every comparison test of the first kind may be deduced from the pair of tests: <; ~ 'Pn PnPn-l * n Pn-l . = A.-I and similarly for other typical forms of series. 4. The right hand sides can be combined to form a single disjunctive criterion, if we introduce a modification, arbitrary in character in so far as it is not necessarily suggested by the general trend of ideas, but otherwise of a simple nature. We see at once, for instance, that the series _ Pn n converge when a > 1 and diverge when a ^ ] . For the first of these scries the proof has just been given; and the second has all its terms less than the first if a > 1, while if a = 1, and hence for all a 2> 1, it is immediately seen to be divergent. The pair of criteria set up in 3. may accordingly be replaced by the following disjunctive criterion: 49 The usual passage from p n direct to log p n , Iog 2 p n , . . . , is again quite an arbitrary step, of course. Theorems 77 and 175, 2 render the step natural, however. Between e. g. p n and log/> n , we could easily introduce intermediary stages, for instance e**^ w hich increases less rapidly than /> n> in fact less rapidly than any fixed positive power of p n , however small its exponent, yet more rapidly than every fixed positive power of log p n , however large its exponent. 42. Systcmatization of the general theory of convergence. 309 and, in all essentials 50 , also by: -i ., (>1 : e with {>! : I ^ 1 : It is remarkable that in the criteria of convergence arising through these transformations, the assumption p n + + oo is no longer necessary at all. It is sufficient that (p n ) should be monotone. In fact, it (p n ) is boun- ded, the convergence of 2(p n p n -j), and hence that of J * - - and -**-*- for arbitrary a > 0, follows from that of (p n ), as (p~ a ) GC H and (cc~ p ) are also bounded sequences. These convergence tests 51 thus possess a special degree of generality, similar to that of Kummers^ cri- terion of the second kind, mentioned below in 7. 5. From this disjunctive criterion as indeed in general from any criterion others may again be deduced by various transformations, though the criteria so obtained can be new only in form. For these transformations we can of course lay down no general rule; new ways may always be found by skill and intuition. This is the reason for the great number of criteria which ultimately remain outside the scope of any given systematization. It is obvious that every inequality may be multiplied by arbitrary positive factors without altering its meaning; similarly we may form the same function F(x) of either member, provided F(x) be monotone increasing (in the stricter sense), in particular we may take log- arithms, roots, etc. of cither side. E. g. the last disjunctive criterion may therefore be put into the form or L : & !*/__ f ^# V P n -Pn-l\ We see at a glance that by this means we obtain a general frame- work for the criteria of the preceding sections which were set up by assuming p n === n or == log n. 50 The equivalence is not complete, i. e. with the same sequence (p n ) as basis, the new criterion is not so effective as the old one; in fact, the divergence of 5? - "" , for instance, may be inferred from the old criterion, but not Pn from the new one 61 Pnngsheim: Math. Ann , Vol. 35, p. 342. 1890 62 Journ. f. d. reine u. an^ew. Math., Vol. 13, p. 78. 1835 11 (o5l) 310 Chapter IX. Series of positive terms. 6. Substantially the same remarks remain valid,, when we sub- stitute "" * for c n and p n p n -i f r d n m me fundamental cri- Pn'Pni terion of the second kind (II), or perform any of the other typical substitutions for c n and d n there. In this way we obtain the most general form of the criteria of the second kind. 1. We may observe (cf. Rem. 4.) that here again, after carrying out a simple transformation, we may so frame the convergence test that it combines with the divergence test to form a single disjunctive criterion. The convergence test requires in the first instance that, for every sufficiently large n, or If here we replace c n by ~ j>~^" ^ ie f rmer inequality reduces to Q . ' pn-1 a n - as p n cancels out, the typical terms of a divergent series automatically appear, so that the convergence test reduces to or . /o Finally, if we take into account the fact that 2 Q d n (Q > 0) diverges with 2d n , the criterion takes the form: n , Now the original criterion is certainly satisfied by the assumption _i a n+\ **> n *t> O - -^ tf -^ v * n a n c n + l It thus appears that in this form slightly less general than the original form of the convergence test, it is absolutely indifferent whether a convergent series or a divergent series is introduced as comparison series. Hence, still more generally, the c n 's and d n 's in the above forms of the criterion may be replaced by any (positive) numbers b n ; thus we may write: Exercises on Chapter IX. 311 This extremely general criterion is due to E. Kummer. On the other hand, ^ ( 181 3) represents a disjunctive criterion of the second kind which immediately follows, as the part relative to divergence is merely a slight trans- formation of (II) All further details will be found in the papers and treatise by A. Pringsheim. The sequences of ideas sketched above can of course lead only to criteria having the nature of comparison tests of the first or second kinds, though all criteria of this character may be developed thereby. The integral test 176 and Ermakoff's test 177 of course could not occur in the considerations of this section, as they do not possess the character in question. Exercises on Chapter IX. 133. Prove in the case of each of the following series that the given indications of convergence or divergence are correct: 2-4... (2n) ' >2 : C, <2 : S>; d) v(__i___lo g -!^^l>) : S; e / ^-j 7*i _L_ i \ /o - _i_ 1 \ /^nrzriT 68 It was given by Kummer as early as 1835 (Journ. f. d. reinc u. angew. Math , Vol. 13, p. 172) though with a restrictive condition which was first re- cognized as superfluous by U.Dmi in 1867. Later it was rediscovered several times and gave rise, as late as 1888, to v.olent contentions on questions of priority. O. Stoh (Vorlesungen liber allgem. Arithmetik, Vol. 1, p. 259) was the first to give the following extremely simple proof, by means of which the criterion was first rendered fully intelligible: Direct proof: The criterion is that from some stage on It follows in particular that the products a n b n diminish monotonely and therefore tend to a definite limit y>0. By 131, (<*& a n + ib n + i) is thus a convergent series of positive terms And as its terms are not less than the corresponding terms of ~a n , this series is also convergent. 312 Chapter X. Series of arbitrary terms. 134* For every fixed p, the expression n \ has a definite limit C p when n > -f- oo , if the summation commences with the first integer for which log p n^>l. 135* For every fixed Q in << g < 1 , the expression has a definite limit y when w -f OO. 136. If #->?, it follows that where p, />', and q denote given natural numbers. 137. If 2d H is divergent, with d n -> , and if the D n 's are its partial suras we have r=l 138. If 2a n has monotonely diminishing terms, it is certainly divergent when p-a>pn ^ for a fixed p and every sufficiently large n. 139. If < d n <C 1 for every n, the two series are convergent, for every Q ^> . 14O. Give a direct proof, without the use of Ermako/f's test and without the help of the integral calculus, of the criterion __ 2a. 2W J <1 : ~^ l>2 : for series of monotonely diminishing terms 141. If the convergence of a series 2a n follows from one of the criteria of the logarithmic scale 164, II, then, as n >-}-oo, [n log n log a n . . . log k n\-a n -> and diminishes monotonely from a certain stage on, whatever the value of the positive integer h may be. Chapter X. Series of arbitrary terms. 43. Tests of convergence for series of arbitrary terms. With series of positive terms, the study of convergence and divergence was capable of systematization to some extent; in the case of series of arbitrary terms, all attempts of this kind have to be abandoned. The reason lies not so much in insufficient de- 43. Tests of convergence for series of arbitrary terms. 313 velopment of the theory, as in the essence of the matter itself. A series of arbitrary terms may- converge, without converging abso- lutely 1 . Indeed this is practically the only case which will interest us here, as the question of absolute convergence reduces, by 85, to the study of a series of positive terms. We therefore need only consider the case in which either the series is actually not absolutely conver- gent or its absolute convergence cannot be demonstrated by any of the previously acquired means. If a series is conditionally conver- gent, however, this convergence is dependent on the mode of succession of the terms as well as on their individual values; any comparison test which we might set up would therefoie have to concern the series as a whole, and not merely its terms individually, as before. This ultimately means that each series has to be examined by itself and we cannot obtain a general method of approach valid for them all. Accordingly we have to be content to establish criteria with a more restricted field of validity. The chief instrument for the purpose is the formula known as Abel's partial summation 9 . // a Q ,a 19 ... and b Q , b^, ... denote 182. arbitrary numbers, and we write <*0 + <*1 H ----- h = A n (" k 0) then for every n ^ and every k^>l, n+fc n+fc Proof. We have by summation from v = n -\- 1 to v n -f- k, the statement at once follows 3 . Supplements. 1. The formula continues to hold when n=^ 1,183, if we put A_i = 0. 1 The case in which the series may be transformed into one with posi- tive terms only, by means of a "finite number of alterations" (v. 82,4) or by a change of sign of all its terms, of course requires no special treatment. 2 Journ f. d. reine u. angew. Math Vol. 1, p. 314. 1826. 1 It is sometimes more convenient to write the formula in the form n + k n + fr-1 ^ a v b v ^- ^ ^v(V-^ + i)~^A* v=n M - n f 1 314 Chapter X. Series of arbitrary terms. 2. If c denotes an arbitrary constant, and A v ' = A v + c, we have also: n + k n + k E a, 6, = E AJ (b v - b l>+l ) - AJ b n+1 + A' n+k . b n+1e+l v-w + l v n+1 for a v = A v A v _i = A v ' A f v _^ Accordingly, in Abel's partial summation we "may" increase or diminish all the A^s by any constant amount. This is equivalent to alter- ing a . Abel's partial summation enables us to deduce a number of tests of convergence for series of the form 2 a v b v almost immediately 4 . In the first place, it provides the following general 184. Theorem. The series 2 ab v certainly converges, if 1) the series 2 A v (b v b v+l ) converges, and 2) lim Aj> - b p+1 exists. p > + x Proof. Abel's partial summation gives for n = 1 : k k Za v b v = 2A v (b v - b, +1 ) + A k b M , v-^O '=0 for every fcJjgO; making &-> + oo, the statement follows, in view of the two hypotheses. The relation just written down shows further that s = s' + I where Sa v b v s, 2 A v (b v b v+1 ) = s', lim A 9 b M = /. In particular, 5 = 5' if, and only if, / = 0. The theorem does not solve the question as to the convergence of the series E a v b v , since it merely reduces it to two new questions; but these are in many cases simpler to treat. The result is in any case a far-reaching one, and it enables us immediately to deduce the following more special criteria, which are comparatively easy to apply. 1. Abel's test 6 . Za v b v is convergent if 2 a v converges and (b n ) is monotone 6 and bounded 7 . 4 We can of course reduce any series to this form, as any number can be expressed as the product of two other numbers. Success in applying the above theorem will depend on the skill with which the terms are so split up. 6 loc. cit. Abel's test provides a sufficient condition to be satisfied by (6 n ), in order that the convergence of 2 a n may involve that of Z a n b n . J. Hadamard (Acta math., Vol. 27, p. 177. 1903) gives necessary and sufficient conditions; cf. E. B. Elliot (Quarterly Journ., Vol. 37, p. 222. 190(5), who gives various refinements. 6 In anticipation of the extension to complex numbers (v. p. 397) it may be em- phasized already that a sequence of numbers assumed to be monotone is necessarily real. 7 In other words: A convergent series "may" be multiplied, term by term, by factors forming a bounded and monotone sequence. Theorem 184 and the criteria deduced from it all deal with the question: By what factors may the terms of a convergent series be multplied so that a convergent series results? And by what factors must the terms of a divergent series be multiplied, so that the resulting series may be convergent? 43. Tests of convergence for series of arbitrary terms. 315 Proof. By hypothesis (A n ) and (b n ), (v. 46), and hence also (A n n+1 ), are convergent. On the other hand, by 131, the series (b v b v+1 ) is convergent, and indeed absolutely convergent, as its terms all have the same sign, in consequence of the monotony of (b n ). It follows, by 87, 2, that the series E A v (b lt b^ +l ) is also convergent, since a convergent sequence is certainly bounded. The two conditions of theorem 184 are accordingly fulfilled and S a v b v is convergent. 2. Dirichlet's test 8 . Za v b v is convergent if 2 a v has bounded partial sums and (b n ) is a monotone null sequence. Proof. By the same reasoning as above, 2 A v (& b v+l ) is con- vergent. Further, as (A n ) is bounded, (A n b n+l ) is a null sequence if (b n ) is, i. e. it is certainly convergent. The two conditions of 184 are again fulfilled. 3. Tests of du Bois-Reymond* and Dedekind a) 2 a v b v is convergent if 2 (b v ^.+i) converges absolutely and a v converges, at least conditionally. Proof. By 87, 2, Z 1 A v - (b v b v+l ) also converges, as (A n ) is cer- tainly bounded. Since further (*o - *i) + (*i ~ *2) + - - + (ft-i - 6) = *o - b n tends to a limit when n -> + GO, so does b n itself; lim A n exists by hypo- thesis, and the existence of lim A n b n+1 follows. b) 2 a v b v is convergent if Z (b v 6, +1 ) converges absolutely and E a v has bounded partial sums, provided b n -> 0. Proof. 2 A v (b v b y+1 ) is again convergent and A n b n+l -> 0. Examples and Applications. 185 1. The convergence of 2 a n involves, by Abel's test, that of E n , 2. J?( 1)" has bounded partial sums. Hence if (6 n ) is a monotone null sequence, 8 Vorlesungen uber Zahlentheorie, l sfc edition, Brunswick 1863, 101. 9 Antnttsprogramm d. Univ. Freiburg, 1871. The designation above adopted for the three tests is rather a conventional one, as all three are substantially due to Abel. For the history of these criteria cf. A. Pringsheim, Math. Ann., Vol. 25, p. 423. 1S85. 10 143 of the work referred to in footnote 8. 316 Chapter X. Series ot aroitrary terms. converges by Dmchlefs test. This is a fresh proof of Leibniz's criterion for series with alternately positive and negative terms (82,5). 3. Given positive integers & , fc 1} k 2 , ... such that 2"( \) kn has bounded partial sums for this the excess of the number of even integers over that of odd integers among the n first exponents fc lf & 2 , . . . as n -> -f oo the series converges, if (& n ) denotes any null sequence. 4. If 2 a n is convergent, the power series 2 a n x n is convergent for 0<jo;<-f-l, since the factors x n form a monotone and bounded sequence. If E a n merely has bounded partial sums, the power series at any rate con- verges for every x such that 0<o;<Cl, since x n then tends to monotoncly. 5. The series ^sinna; and J cos.no; have bounded partial sums, the first for every (fixed) real x and the second for every (fixed) real x not a multiple of 2ji. This follows from the following elementary but important formula, valid ll for every x 4= 2 k iri x / x\ sin n sin f a. + (n -f- 1) ~J sin (a + x) -f sin (a -f 2 x) -f- --- H sin (a -f n x) = -- . sin | The proof of the formula is given in 201. For a = 0, we get sin n - sin (n -f- 1) sin x -f sin 2 a; -f- -f- sin n x = --- , (x =j= 2 fc rc) sin - , . JT and for a = - , a; a; sinn -cos ( n +l)-- cos a; + cos 2 cc + -f cos w a; = - , (x -]- 2 A n) . sin | From this the boundedness of the partial sums can be inferred at once. Thus if 2(b n & n +i) converges absolutely and & /t ->0, we conclude from the criterion 3b that ^T b n sin n x converges for every x , 2b n cos n x converges for every x =f- 2 & ^r . In particular 12 , this is the case when b n diminishes monotonely to 0. 6. If the b n 's are positive, and if we may write where 3 > o and (/? n ) is bounded, then 2"(-- l) n b n converges if, and only t/ f a > 0. In fact, if a > 0, it follows from these hypotheses that --!<; 1 from some stage on, i. e. (6 n ) decreases monotonely, and the convergence of the series in question is therefore secured by 2., if we can show that 6 n ->0. The proof of this is similar to that of the parallel fact in 17O, 1 : Jf < a' < a, we have for every sufficiently large v t say v>w, 11 For x=2k7t, the sum has obviously the value n sin a, for all n's. 18 Malmsten, C. /.: Nova acta Upsaliensis (2), Vol. 12, p. 255. 1844. 43. Tests of convergence for series of arbitrary terms. 317 Writing down this inequality for v = m, m-f-1, ..., n 1 and multiplying together, we obtain From the divergence of the harmonic series, it follows as in 170, 1 that 6 n ->0. In the case a < 0, b n must for similar reasons increase monotonely from some stage on, so that 2 ( l) n b n certainly cannot converge. Finally, when a = 0, we deduce in precisely the same way as on p. 289, that b n cannot tend to and the series therefore cannot converge. 7. If a series of the form ,57^ such series are known as Dirichlet ** YI X series; we shall investigate them in more detail later on ( 58, A) is con- vergent for a particular value of x, say x = x , it also converges for every x>x 09 for f ) is a monotone null sequence. This simple application of \n x ~ x J Abel's test, by reasoning quite similar to that employed for power series (93), leads to the theorem: Every series of the form ^ ~ possesses a definite abscissa of convergence JL with the property that the series converges whenever x > Jl and diverges whenever x<Ji. (For further details, v. 58, A.) General Remarks. 186. 1. We have already mentioned the fact that the magnitude of the indiv- idual term in an arbitrary scries is not conclusive with regard to convergence. In particular, two series 2 a n and Zb n , whose terms are asymptotically equal, i. e. such that * - 1 , need not exhibit the same behaviour as regards con- vergence (cf. 7O, 4). Thus e. g. for we have b n ~~ log n ~* But Sb n is convergent and S a n divergent, since 2(a n b n ) diverges by 79,2. 2. // the series ~ a n is non-absolutely convergent^ (cf. p. 136, footnote 9), its positive and it* negative terms, taken separately, form two divergent series. More precisely, let p n a n when a n > 0, and = when a n < 0, and similarly let q n = a n when a n < 0, and =0 when a w ^>0. 18 The two series 2p n and 2 q n are scries of positive terms, the first containing only the positive terms of 2 a n and the second only the absolute values of the negative terms of 2a nj in either case with the places unchanged, while their other terms are all 0. Both these series are divergent. In fact, as every partial sum of 2 a n is the difference of two suitable partial sums of 2p H and 2 q n , it follows at once that if 2 p n and 2 q n were both convergent, so would 2"|a w | be (by 70), contrary to hypothesis; and if the one were convergent, the other divergent, the partial sums of 2 a n Thus i> - ' *" ' 1 11US p n - it* (051) 318 Chapter X. Series of arbitrary terms. would tend to - oo or -f oo (according as 2 p n 01 2 4n is assumed convergent), which is again contrary to hypothesis. 3. By the preceding remark, a conditionally convergent series, or rather the sequence formed by its partial sums, is exhibited as the difference of two monotone increasing sequences of numbers tending to infinity 14 . As regards the rapidity with which these increase, we may easily establish the following Theorem. The partial sums of 2 p n and 2 q n are asymptotically equal. In fact, we have since the numerator in the latter ratio remains bounded, while the denominator increases to -J-CX) with n t this ratio tends to 0, which proves the result. 4. The relative frequency of positive and negative terms in a conditionally convergent series 2 a n for which \a n \ diminishes monotonely is subject to the following elegant theorem, due to E. Cesaro: The limit, if it exists, of the p ratio - n of P M the number o/ positive terms } to Q M the number of negative terms a vt Qn for v<w, is necessarily 1 44. Rearrangement of conditionally convergent series. The fundamental distinction between absolutely and non-absolutely convergent series has already been made clear in 89, 2. This is, that the behaviour of non-absolutely convergent series depends essentially on the order of the terms in the series, so that for these series the commutative law of addition no longer holds. The proof consisted in showing that a non-absolutely convergent series could, by a mere re- arrangement in the order of its terms, be transformed into a divergent series. This result may now be considerably elaborated. In fact it may be shewn that by a suitable rearrangement any prescribed behav- iour, as regards convergence or divergence, may be induced. The theorem which we obtain is 187. Riemann's rearrangement theorem. // 2 a n is a conditionally convergent series, we may, by a suitable rearrangement (v. 27, 3), de- duce a series 2a^ with any one of the following properties: 14 It is best to avoid, as being far too superficial in character, the mode of expression which may be found in some writings: "the sum of a condition- ally convergent series is given in the form CO CO." Rom. Ace. Lincei Rend. (4), Vol. 4, p. 133 1888. Cf. a Note by G. H. Hardy, Messenger of Math. (2), Vol. 41, p. 17. 1911, and one by H.Radt macher. Math. Zeitschr., Vol 11, pp. 276288. 1921. 44. Rearrangement of conditionally convergent series. 319 a) to converge to an arbitrary 16 prescribed sum s' ; b) to diverge to -f- oo or to oo ; c) to exhibit as upper and lower limits of its partial sums two arbitrary numbers p and x, with ju^>x. Proof. It suffices to prove c), since a) and b) are particular cases of c), the former for = p, = s' and the latter for x = p, = -f- oo or = oo. To prove c), let (x n ) be any sequence tending to x and (/* n ) any sequence tending to p, y with jji n > x n and 17 ^ > 0. Let us denote by p^ 3 , . . . the terms in 2 a n E= a 1 ^- # 2 H ---- which are ^ 0, in the order in which they occur, and by q^ 9 q^, . . . the absolute values of those which are < 0, again in their proper order, thus slightly modifying the definition in 186, 2. The series 2p and 2 q n only differ from those in 186, 2 by the absence of a number of zero terms, and are accordingly both divergent, with posi- tive terms which tend to 0. We proceed to show that a series of the type Pl + P* ---- h Pm l ?i ft ----- ?*, + Pm^l H ----- h fe+i ----- fc + #+i H ---- will satisfy all the requirements. Such a series is clearly a re- arrangement of the given series, and is indeed one which leaves un- altered the order of the positive terms relatively to one another and that of the negative terms relatively to one another. Let us choose the indices m^ < m 3 -. ..., k < k 2 <C . . ., in the above series, so that: 1) the partial sum whose last term is p mi has a value > // x , while that ending one term earlier is ^ /^ ; 2) the partial sum whose last term is q^ has a value < ^, while that ending one term earlier is ^> x x ; 3) the partial sum whose last term is p m ^ has a value > yu , while that ending one term earlier is <^ /v 2 ; 16 Riemann.B.: Abb. d. Ges. d. Wiss. z. Gottingcn, Vol. 13, p. 97. 186668. The statements b) and c) are obvious supplementary propositions. 17 This is clearly possible in any number of ways. In fact, if * = /* with a finite value s', say, take * = sf and ft n = s'H , taking ^. even larger, n n if necessary. If x = ^ = -f OO ( oo), take * w = n ( n) and // n = x w + 2. If, finally, *<ft, take any (x n ) and (fi w ) tending to and p\ from some stage on, # n <C^Mn, and by a finite number of alterations, we can arrange that this may be the case from the beginning, and also that ^^O. 320 Chapter X. Series of arbitrary terms. 4) the partial sum whose last term is q kl has a value < x 2 , while that ending one term earlier is ^ x 2 ; and so on. This can always be arranged; for by taking a sufficient number of positive terms, the partial sum may be made as large as we please, and by allowing a sufficient number of negative ones to follow, the partial sum may again be depressed below any assigned value. On the other hand, at least one term must be taken at each stage, since x n < /z n ; so every term of the original series really does occur in the new series. Let H a n ' denote the definite rearrangement of E a n so obtained; the partial sums of S a n ' have the prescribed upper and lower limits. In fact, if for brevity we denote by r lf r 2 , . . . , the partial sums whose last terms are /> Wl , /> m2 , . . . and by cr l9 o- 2 , . . . , those whose last terms arc fe fc, , we have Since p n -> and q n -> 0, it follows that a, -> x and r v -> /z, so that x and /* certainly represent values of accumulation of the partial sums of 27 a n . Now a partial sum s n f of 2 a n \ which is neither a a v nor a T,,, has necessarily a value between those of two successive partial sums of this special type; hence s n ' can have no value of accumulation outside the interval x . . . /i, (or different from the common value of x and IJL if these coincide). In other words, /x and x are themselves the upper and the lower limit of the partial sums, q. e. d. Various researches of an analogous nature were started in different directions as a consequence of this theorem. M. Ohm 18 and O. Schlomilch 19 investigated the effect of rearrangement on the special series 1 ^ + - -+ ..., in par- ticular the case in which p positive terms are followed by q negative terms throughout (cf. Exercise 148). A. Pringsheim 20 was the first, however, to aim at general results for the case in which the relative frequency of the positive and negative terms in a conditionally convergent series is modified according to definite prescribed rules. E. Borel 21 investigated the opposite problem, as to what rearrangements in a con- ditionally convergent series do not alter its sum. Later, W. Sierpinski 22 showed that if 27 a n = s converges conditionally and s' < s t the series can be made to have the sum s' by rearranging only the positive terms in the series, leaving all the negative terms with unaltered place and order > while similarly it can be made to have any sum s" > s by rearranging only the negative terms. (The proof is not so simple.) 45. Multiplication of conditionally convergent series. We showed in the preceding section, thus completing the con- siderations of 89, 2, that the commutative law of addition no longer holds for series which converge only conditionally. We have also seen 18 Antrittsprogramm, Berlin, 1839. 19 Zeitschr. f. Math. u. Phys., Vol. 18, p. 520. 1873. 20 Math. Ann., Vol. 22, p. 455. 1883. 21 Bulletin des sciences mathcm. (2), Vol. 14, p. 97. 1890. 22 Bull, internat. A'c. Sciences Cracovie, p. 149. 1911. 45. Multiplication of conditionally convergent series. 321 already (end of 17), in an example due to Cauchy, that the dis- tributive law does not in general subsist, so that the product of two such series 2 a n and 2b n may no longer be formed according to the elementary rules. The question remained unsolved, however, whether the product series Sc n (with c n = a b n + a^ 6 W _ 1 -j (- a n 6 ) might not continue to converge under less stringent conditions for 2 a n = A and 2b n = B, and to have the sum A-B. In 17, it was required that both Za n and 2'6 M should converge absolutely. In this connection, we have first the Theorem oiMertens 23 . If at least one of the two convergent series 188. Z a n = A and 2 b n = B converges absolutely, E c n converges and = A B. Proof. We have only to show that, with increasing n, the partial sums C n = <0 + c l -!-...+* = *o h o + K *i + <*i b o) + - + ( a o b n + <*i b n -i + + a n b ) tend to A B as limit. We may assume that Z a n is, of the two series, the one that converges absolutely. If we denote by A n the partial sums of 27 # n , by B n those of H b nj we have C n = <*o-B n + <* 1 B n _ 1 + + B , or, if we put B n = B -\- fl n , = *n B + ( /?+ 1 /.-!+-+ A)' Since ^4 n -B >-yl -5, it only remains to show lhat when 2a n is absolutely convergent and /? n *(), the expressions w = .^o + .-iA+- + o/ form a null sequence. But this is an immediate consequence of 44, 9 b; we have only to put x n = f) n and y n = a n there. Thus the theorem is proved. Finally, we shall answer the question whether the product series 2c n , if convergent, necessarily has the sum A-B. The answer is in the affirmative, as the following theorem shows: Theorem of Abel**. If the three series 2a n , 2b n and 189. 2c n = 2(a b n -}-- + n & ) are convergent, and A, B, and C are their sums, we have A-B C. 1. Proof. The theorem follows immediately from Abel's limit theorem (10O) and was first proved by Abel in this way. If we write S* * n = /"x (*)> 2b n x = /; (x), 2c n x n = f s (*), i J. f. d. reine u. angew. Math., Vol. 79, p 182. 1875. An extension was given by T. /. Stieltjes (Nouv. Annales (3), Vol.6, p. 210. 1887). 94 J. f. d. reine u. angew. Math., Vol. 1, p. 318. 1826. 322 Chapter X. Series of arbitrary terms. these three power series (cf. 185, 4) certainly converge absolutely for <^ a; < 1, and for these values of x, the relation (a) A (*)/;(*) = /;(*) holds. The assumed convergence of 2a n , 2b n and 2c n implies, by Abel's limit theorem 100, that each of the three functions tends to a limit when a? * + 1 from the left; and fi(x)-+A=2a n , f,(x)-+B = Sb n , f^(x)-+C = 2c n . Since the relation (a) holds for all the values of x concerned, it follows (by 19, Theorem l) that it must hold in the limit: We may also dispense with the use of functions and adopt the following 2. Proof due to Cesdro**. It was shown above that From this it follows that Dividing both sides of this equality by n -f- 1 and letting n + -f- oo, we obtain C as limit on the left hand side (by 43,2) and A-B as limit on the right (by 44, 9 a). Hence A-B = C, q. e. d. In consequence of this interesting theorem, with which we shall again be concerned later on, any further elaboration of the question of multiplication of series has only to deal with the problem whether the series 2 c n converges. Into these investigations we do not, however, propose to enter 36 . Examples and Applications. 1. It follows from !L = ^^ = 1 - J. + -L _ I + . . ., by the pre- * n-rO ^ n ~>~ L * f) ' cedingf theorem, that provided the series thus obtained converges. 26 Bull, des sciences math. (2), Vol. 14, p. 114. 1890. 26 Theorems of the kind in question have been proved by A. Pringsheim (Math. Ann., Vol.21, p. 340. 1883), and in connection with the latter's work, by A. Voss (ibid. Vol. 24, p. 42. 1884) and F. Cajori (Bull, of the Americ. Math. Soc., Vol. 8, p. 231. 1901-2 and Vol. 9, p. 188. 1902-3). Cf. also 66 of A. Prings heim's treatise, Vorlesungen Uber Zahlen- und Funktionenlehre (Leipzig- 1916), to which we have already referred more than once. G. H. Hardy (Proc. Lon- don Math. Soc. (2), vol. 6, p. 410, 1908) has proved a particularly elegant example of a related group of much more fundamental theorems. 45. Multiplication of conditionally convergent series. 323 Now (2 p + 1) (2 n + 1 - 2 ) 2ln+l)V2 + 12n so that the generic terra of the new series has the value , , , r _1_\ w-M,/ ' _ n+1 V 3 2 Since ^ - t - tends monotoncly to zero, so docs its arithmetic mean 4 W -f" 1 and the new series therefore does converge by Leibniz's test 82, 5 We thus have as 2. In a precisely similar manner, we deduce (v. ISO), by squaring- the series log 2 = 1 - + --- h , 3. The result obtained in 1. provides a fresh mode of approach to the 00 1 7T 3 equation J>!\ f > = --, which has occupied us repeatedly before now (v. 136 *=!*" b and 15ft) To see this, we first prove the following- Theorem. Let (a ot a lt 2 , . . .) fo a monotone sequence of positive numbers. for which 2 a n 2 ts convergent. Then the series 1- J}( 1)" = *; 2 ^ a n a n+f = S J/f p=1.2,..., n~0 n=0 anc2 3. (-1) P =J, converge, with (c) J> M * = s*-2J. n=o Proof. Since -S 1 ^ 9 converges, a->0; accordingly the series 1 con- verges by Leibniz's test. As a n a n + p < a w s for every ^ ^ 1, and ^"a n a converges, the series 2 are also convergent for >1. Further, as a n + p + l -< a n + p9 we have d p + i<_d p . The series 3 will accordingly converge if cJ^-^O. Now given e ^> 0, we can choose m so that aji . 1 -j- aj* , a -J- - - ^-pr-: for every suffi- ra ' 77 T" A l T a 2 ciently large p, we shall then have 8 S p < a O a /> 4- ^*i a /; + i + ' -f a m a p + m + "2" < a p ( a O "I" a i H h O -f -g- < - Hence o p and the series 3 also converges. Let us now form the array a, a -f- a/ 2 a, a, -f ^ a, H . 324 Chapter X. Series of arbitrary terms. and let S n denote the sum of the products ^ a l a ft for which A and fi are <j n. These obviously fill up a square in the upper left hand corner of the array, and S n = (a -a 1 + ---- + (_!) )_,!. On the other hand, the sum of all the (primary) diagonals which contain at least one product a^ a^ belonging: to that square, is clearly r-O Hence, to obtain (cj, it now suffices to prove that T n S n *0. By writing out the above array in a more detailed fashion, we see, moreover, that (- l) n (T n - S n ) == 2 [a a a n +1 + 3 a n + 2 -f ---- ] - 2 [a a a n hl -f- 3 <? + 3 -\ ---- ] + 1 + 4 a n f-2 H ---- ] - I ---- - 1 -2[a j ,fl n + 1 + a^ t-i This we write for brevity - a, - a + ,- + ... + (_ l)--i + (- 1)" /? n , and as a, ^ r<r.j > > a n ^ 0, we have (of. 81 c, 1^) \ T n~ ^[^^-r-^^^-f/?!,; thus, as was asserted, T n 5 7l ->0 and therefore ^ M 2 s 3 2 /f . 4. If, in 3., we now take a n = - -- -, the hypotheses are obviously all 2i 11 -f- 1 fulfilled, and we have But in this case, we have, by 133,1, 00 1 * y^ # f for every p > 1 , and hence _ __ -_, yi , ..... n=0^ 2w + 1 ) a 16 n-0 "+ 1 V 3^ + 2n By the equality (a) proved in 1., the right hand side = -=-. By the method o op 1 ^2 used to deduce 137 from 136, the equality - - = follows at once ^=-l ^ b The fresh proof thus obtained for this relation may be regarded as the most elementary of all known proofs, since it borrows nothing from the theory of functions except the Leibniz series 122. The main idea of the proof goes back to Nicolaus Bernoulli 21 . Exercises on Chapter X. 142. Determine the behaviour of the following- series: n=l ' sin \ 87 Comment. Ac. Imp scient. Petropolitanae, Vol. X, p. 19. 1738. Exercises on Chapter X. 325 e) 27(-l)" sin-?-, f) 27sini f g) 2&\n(n*x), h) 2 sin (w! nx), (~ 1)n s *" 2 "* 4- + '" + -, m) 2 n It n J n In the last series, (ct n ) is a monotone null sequence. The series g) does not converge unless x-kn\ the series h) converges for all rational values of x, also e. g. for 2 k v = e, = (2fc-f-l)0, = , = sin 1, = cos 1, and for 1 1_ !_ 1 2__ 1_ 1 1 X "24! 5! + 2~6! 7 ! "*" 2 8 ! "*""" and many other special values of a?. Indicate values of x for which it cer- tainly cli \cTges. *. J7 L-qraV-i + F+ aU ~ ^] " log 2 for every x ;> . 144. If (wa w ) and 2" w (a n a n + 1 ) converge, the series Z a n also con- verges. 145. a) If 2a n and 2\b n b n + l \ both converge, or b), if 2a n has bounded partial sums, 2 1 /> b n + 1 1 converges and 6 n -0, then for every integer p 2> 1 the series 2a H b H * is convergent. 146. The conditions of the test 184, 3 are in a certain sense necessary, as well as sufficient, for the convergence of 2a n b n : If it be required that for a given (&), 2a n b n always converges with 2a nt the necessary and sufficient condition is that 2\b n b n + l \ should converge. Show also that it makes little difference in this connection whether we require that ^\b n b n + l \ con- verges or merely that (&) is monotone. 147. If 2a n converges, and if p n increases monotonely to -f-oo in such a way that 2p n ~ l is divergent, we have n 148. Let a n tend to monotonely, and assume that hm n a n exists. If 00 we write J? ( 1) W M = 5 > an ^ now rearrange this series (cf. Ex. 51) so as to n=o have alternately p positive and q negative terms: a o-i-^ + -"+ a 2i-a--0i-*3 ----- a a ?- !+**+. the sum s' of the new series satisfies the relation s' = s -f hm (n a n )-log ?-. * 9 140. A necessary and sufficient condition for the convergence of the product series v CB = l'(a 6 B + a l 6, 1 _ 1 +... + a B 6 ) of two convergent series -Ta n , 2b nt is that the numbers fib=-</M& + ^-l + --- + ^-r4.|) v=l should form a null sequence. 326 Chapter XL Series of variable terms. 150. If (a n ) and (b n ) are monotone sequences with limit 0, the Cauchy's product series of 2( l) n a n and 2( l) n b n is convergent if, and only if, the numbers o n = a n (b Q -f b t -f- - + b n ) and r n = b n (a -f- a t ^ ----- j- a n ) also form a null sequence. 151. The two series ^~^" and -^7 "/si ' ' >> >0> may be multiplied together by Cauchy's rule if, and only if, a-J-^>l. 152. If (#) and (& n ) are monotone null sequences, Cawc/ty's product of the series 2( !)" and J( l)"& n certainly converges if 2a H b n converges. A necessary and sutficient condition for the convergence of the product series is that 2(a n b n ) l '*~ e should converge for every 153. If, for every sufficiently large n t we can write a n = n tt '.(logn)' (log, n)" (log r n)' , 6 n = w^.(logw/' (Iog 3 nA (log, nf* , and if 2b n converges, we have, provided a n is not equal to b n for every n t -f a, &_, r-0 Chapter XI. Series of variable terms (Sequences of functions). 46. Uniform convergence. Thus far, we have almost exclusively taken into consideration series whose terms were given (constant) numbers. It was only in particularly simple cases that the value of the terms depended on the choice of a definite quantity, or variable. Such was the case e. g. when we were considering the geometric series 2 a n or the harmonic series y 1 ; their behaviour was dependent on the choice of a or of a. A more n a general example is that of the power series 2 a n x n , where the number x had to be given, before we could attack the problem of its con- vergence or divergence. This type of case will now be generalized in the following obvious way: we shall consider series whose terms depend in any manner on a variable x, i. e. are functions of this variable. We accordingly denote these terms by f n (x) and consider series of the form 2f n (x). A function of x, in the general case, is defined only for certain values of x (v. 19, Def. 1); for our purposes, it will be sufficient to assume that the functions f n (x) are defined in one or more (open or closed) intervals For the given series to have a meaning for any value 46. Uniform convergence. 327 of x at all, we have to require that at least one point x belongs to the intervals of definition of all the functions f n (x). We shall, however, at once lay down the condition that there exists at least one interval, in which all the functions f n (x) are simultaneously defined. For every particular x in this interval, the terms of the series 2f n (x) arc in any case all determinate numbers, and the question of its convergence can be raised. We shall now assume further that an interval / (possibly smaller than the former) exists, for every point of which the series f n (x) is found to converge. Definition 1 . An interval J will be called an interval of conver- 190. of the series 2f n (x) if, at every one of its points (including one, both, or neither of its endpoints), all the functions f n (x) are defined and the series converges. Examples and Illustrations. 1. For the geometric series 2 x n , the interval 1 < a; <C + 1 is an interval of convergence, and no larger interval of convergence exists outside it. 2. A power series ~ a n (x x ) n , provided it converges at one point at least, other than x 09 always possesses an interval of convergence of the form (x r) . . . (Xfi-i-r), inclusive or exclusive of one or both endpoints. When r is properly chosen, no further interval of convergence exists outside that one. 1 8. The harmonic series 2j has as interval of convergence the semi- n* axis x > 1 , with no further interval of convergence outside it. 4. As a series is no more than a symbolic expression for a certain se- quence of numbers, so the series 2f n (x) represents no more than a different symbolic form for a sequence of functions) namely that of its partial sums In principle, it is therefore immaterial whether the terms of the series or its partial sums are assigned, as each set determines tht other uniquely. Thus, in principle, it also does not matter whether we speak of infinite series of variable terms or of sequences of functions. We shall accordingly state our definitions and theorems only for the case of series and leave it to the student to formulate them for the case of sequences of functions*. 5. For the series r+ j 1 For the case of complex numbers and functions, we have here to substitute throughout the word region for the word interval and boundary points of the region for endpoints of the interval. With this modification, the sign o has the same sig- nificance in this chapter as previously. 2 Occasionally, however, the definitions and theorems will also be applied to sequences of functions. 328 Chapter XI. Series of variable terms. we have The series converges for every real x. Clearly, indeed, we have a) s H (x)-+0, if |*,<1, b) s n (x) -> 1 , if | x | > 1 , and c) s*(x)-+l, if |0| = 1. 6. On the other hand, defines a series with an infinity of separate intervals of convergence; foi lim s n (x) obviously exists if, and only if, -<T <C sin x < 75- , i.e. if or ' ^ or if a; lies in an interval deduced from these by a displacement through an integral multiple of 2vt. The sum of the seties = throughout the interior of the interval and = 1 at the included endpoint. - , . sin 2 x sin 3 a: o *- r r 7. Ihe series sin x ~\ -- -^ -- 1 -- -- h converges, by 185, 5. for every fj O cos 2 a; , cos 3 a: , . . real x\ the series cosxH --- ^ -- 1 -- -- 1 ---- converges for every real 6 O X =f= 2kyt. If a given series of the form 2 f n (x) is convergent in a deter- minate interval /, there corresponds to every point of / a perfectly definite value of the sum of the series. This sum accordingly ( 19, Def. l) is itself a function of x, which is defined or represented by the series. When the latter function is the chief centre of interest, it is also said to be expanded in the series in question. In this sense, we write n=0 In the case of power series and of the functions they represent (v. Chapters V and VI), these ideas are already familiar to us. The most important question to be solved, when a series of variable terms is given, will usually be whether, and to what extent, properties belonging to all the functions f n (%), i. e. to the terms of the given series, are transferred to its sum. Even the simple examples given above show that this need not be the case for any of the properties which are of particular interest in the case of functions. The geometric series shows that all the func- tions f n (x) may be bounded, without F (x) being so; the power series for sin a;, x > 0, shows that every f n (x) may be monotone, without F(x) being so; example 5 shows that every f n (x) may be continuous, 46. Uniform convergence. 329 without F(x) being so, and the same example illustrates the corres- ponding fact for differentiability. It is easy to construct an example showing that the property of integrability may also disappear. For instance, let { = 1 for every rational x expressible as a fraction with denominator (positive and) < n, = for every other x . Then s n (x), for each n, and consequently f n (x), for each n, is intc- grable over any bounded interval, as it has only a finite number of discon- tinuities in such an interval (cf. 19, theorem 13) Also lim s n (x) = F (x) exists for every x. In fact, if x is rational, say = (?>0, p and q prime to one another), we have, for every n>q, s n (x) = l and hence F(x) = l. If, on the other hand, x is irrational, s n (a;) = for every n and so F(o;) = 0. Thus ~fn ( x ) = lim s n (x) defines the function = 1 for a rational x, = for an irrational x. This function is not integrable, for it is discontinuous 3 for every x. Even by these few examples, we are led to see that a quite new category of problems arises with the consideration of series of variable terms. We have to investigate under what supplementary con- ditions this or the other property of the terms f n (x) is transferred to the sum F(x). It is clear from the examples cited that the mere fact of convergence does not secure this, the cause must reside in the mode of convergence. A concept of the greatest importance in this respect is that known as uniform convergence of a series 2 f n (x) in one of its intervals of convergence or in part of such an interval. This idea is easy to explain, but its underlying nature is not so readily grasped. We shall therefore first illustrate the matter somewhat intuitively, before proceeding to the abstract formulation: CO Let 2 f n (x) converge, and have for sum F(x), in an interval/, a <x<b- n=0 we shall speak of the graph of the function y = s n (x) = f (x) -\ h f n (x) as being the nU* curve of approximation and of the graph of the function y = F(x) 8 We may modify this definition a little by taking s n (x) = 1 for all rational xs whose denominators are factors of n^ and =0 elsewhere; the rational a?'s in question comprise, for each n, a definite number of other values besides the integers <w used above. We then obtain as lims n (#) the same function F(x) as above. In this case, however, both s n (x) and F(x) may be represent- ed in terms of a closed expression, by the usual means; in fact, we have $n (x) = lim (cos 2 nl nx) k t and therefore F(x) = lim [ lim (cos 2 n! ytx) k }. This curious example of a function, discontinuous everywhere, yet obtainable by a repeated passage to the limit from continuous functions, is due to Dirichlet. 330 Chapter XI. Series of variable terms. 00 as the limiting curve. The fact of the convergence of f n (x) to F (x) in J n=o then appears to imply that for increasing- n, the curves of approximation lie closer and closer to the limiting curve. This, however, is only a very imper- fect description of what actually occurs. In fact, the convergence in / implies only, in the first instance, that at each individual point there is convergence; all we can say, to begin with, is therefore that when any definite abscissa x is singled out (and kept fixed) the corresponding ordinates of the curves oi approximation approach, as n increases, the ordinate of the limiting curve for the same abscissa. There is no reason why the curve y = s n (x) , as a whole, should lie closer and closer to the limiting curve. This statement sounds rather paradoxical, but an example will immediately make it clear. The series whose partial sums for n= 1, 2, ... have the values certainly converges in the interval 1 < x < 2 . In fact, in that interval, The limiting curve is therefore the stretch 1 < a; < 2 on the axis of x. The n** curve of approximation lies above this stretch and, by the above inequality, at a distance of less than from the limiting curve, throughout the whole of the interval 1 < x < 2 . For large n's, the distance all along the curve is therefore very small. In this case, therefore, matters are much as we should expect; the position is entirely altered if we consider the same series in the interval < x < 1 . We still have lim s n (x) = at every point of this interval 4 , so that the limiting curve is the corresponding poition of the a;- axis. But in this case the n** 1 approximation curve no longer lies close to the limiting curve through- out the interval, for any n (however large). For #== , we have always s n (a:) = , so that, for every n, the approximation curve in the interval from t to 1 has a hump of height ~ ! ! The graph of the curve y = S 4 (x) has the 2 following appearance: Fig. 4. 4 In fact, for x > we have < s n (x) < as before, i. e. < e for n x every n > ; for x , s n (x) = even permanently. X The curve y 46. Uniform convergence. 331 ), however, corresponds more nearly to the following graph: Fig. 5. For larger n's, the hump in question without diminishing in height be- comes compressed nearer and nearer to the ordinate-axis. The approximation curve springs more and more steeply upwards 5 from the origin to the height , Ci which it attains for x , only to drop down again almost as rapidly to n within a very small distance of the a; -axis. The beginner, to whom this phenomenon will appear very odd, should take care to get it quite clear in his mind that the ordmates of the approxima- tion curves do nevertheless, for every fixed x, ultimately shrink up to the point on the tf-axis, so that we do have, for every fixed x, lim s n (x) = 0* If x is given a fixed value (however small), the disturbing hump of the curve y=s n (x) will ultimately, i. e. for sufficiently large w's, be situated entirely to the left of the ordinate through x (though still to the right of the y-axis) and on this ordinate the curve will again have already dropped very close to the a;- axis*. Therefore the convergence of our series will be called uniform in the interval 1 < x < 2 , but not in the interval < x < 1 . We now proceed to the abstract formulation: Suppose 2f n (x) possesses an interval of convergence /; it is convergent for every indiv- idual point of /, for instance at X = X Q \ this means that if we write F(x) = s w (#) + r n ( x ] and assume > arbitrarily given, there is a number n Q such that, for every n > n , Of course the number n Q , as was already emphasized (v. 10, rem. 3), de- pends on the choice of e. But n Q now depends on the choice of X Q also. In fact for some points of / the series will in general converge more 6 At the origin, its slope is s n '(Q) = n. 8 If we take, say, x = y^r~ and n = 1000000, the abscissa of the highest point of the hump is ffinnftflft' and at our point * the curve has already dropped to a height < . 332 Chapter XI. Series of variable terms. rapidly than for others 7 . By analogy with 10, 3, we shall therefore write w = w (e,a: ); or more simply, dispensing with the index and with the special emphasis on the dependence on e, we shall say: Given e > and given x in the interval /, a number n (x) can always be assigned, such that for every n > n (x) , If we now assume n(x) still for the definite given e chosen, say as an integer, as small as possible, its value is then uniquely defined by the value of x; as such it represents a function of #. In a certain sense, its value may be considered as a measure of the rapid- ity of convergence of the series at the point x. We now define as follows: 191. Definition of uniform convergence (1 st form). The series 2f n (%) convergent in the interval f, is said to be uniformly convergent in the sub-interval ]' of /, if the function n(x) defined above is bounded in /', for each value 8 of e. Supposing we then have n (x) < N in / this N will of course depend on the choice of c, like the numbers n(x) themselves we may also say: 2 nd (principal) form of the definition. A series 2f n (x), con- vergent in the interval J, is said to be uniformly convergent in a sub-interval /' of /, if, given e, a single number N = N(e) can be assigned independently of x, such that not only (as formerly) for every n>N, but also for every x in /'. We also say that the remainders r n (x) tend uniformly to in /'. Illustrations and Examples. 1. Uniformity of convergence invariably concerns a whole interval, nevei an isolated point 9 . 2. A series 2 f n (x) convergent in an interval / does not necessarily con- verge uniformly in any sub-interval of /. 3. If the power series 2 a n (x X ) n has the positive radius r and if the series is uniformly convergent in the closed sub-interval /' of 7 The student should compare, for instance, the rapidity of convergence of the geometric series 2x n (i. e. the rapidity with which the remainder diminishes 1 99 as n increases) for the values 3 = -^ and 05 = ^^^. IUU 1UO 8 If, that is to say, the above-mentioned measure of the rapidity of con- vergence evinces no unduly great irregularities in the interval /'. In par- ticular cases /' may of course consist of the complete interval /. 9 More generally, it may have reference to sets of points more than finite in number. 46. Uniform convergence. 333 its interval of convergence, defined by < x x < + Q. In fact , as the point x *= #Q-}- li es m tne interior of the interval of convergence of the power series, the latter is absolutely convergent at that point. But if 2a n Q n converges absolutely, we can, given e ;> 0, choose N=N(e) so that for every n> N I +i \-e n+ * -f I *+2 1 -e n - M -h <. Also, since |a5 x Q \<Q for every a; in /', we have Thus for n ^> N, we certainly have | r n (x) \ O> whatever the position of x in /' may be . The result we have obtained is as follows o Theorem. A power series 2a n (x~x ) n of positive radius r converges uni- formly in every sub-interval of the form \ x X \ < Q < r of its interval of con- vergence. 4. The above example enables us to make ourselves understood, if we formulate the definition of uniform convergence a little more loosely, as follows: 2 f n (x) is said to be uniformly convergent in J', if it is possible to make a statement about the value of the remainder, in the form "\ r n (x) \ < *", valid for all positions of x simultaneously. 5. The series 2 - 5 is uniformly convergent for every value of x\ n=l * for, whatever the position of x may be, ~ whence the rest may be inferred by 4. (>. The geometric series is not uniformly convergent in the whole interval of convergence - 1 < x < -|- 1. For however large N may be chosen, we can always find an r n (x) with n "> N and < x < 1, for which e. g. r n (x) > 1. If, for instance, we choose any fixed n > N t then as x -> 1 we have x n+i Hence r n (x) > 1 for all x in a definite interval of the form x < x < 1. 7. The above clears up the meaning of the statement : Sf n (x) is not uni- formly convergent in a portion J' of its interval of convergence. A special value of e, say the value e<j > 0, exists, such that an index n greater than any assigned N may be found, so that the inequality | r n (x) | < e is not satisfied for some suit- ably chosen x in J' . 8. With reference to the curves of approximation y = s n (x) t our definition clearly implies that, with increasing n t the curve should lie arbitrarily close to the limiting curve throughout the portion which lies above J'. If, for any given e > 0, we draw the two curves y = F (x) e, the approximation curves y = s n (x) will ultimately, for every sufficiently large n, come to he entirely within the strip bounded by the two curves. 334 Chapter XI. Series of variable terms. 9. The distinction between uniform and non-uniform convergence, and the great significance of the former in the theory of infinite series, were first re- cognized (almost simultaneously) by Ph. L. v. Seidel (Abh. d. Mimch. Akad. r p. 383, 1848) and by G. G. Stokes (Transactions of the Cambridge Phil. Soc., Vol. 8, p. 533. 1848). It appears, however, from a paper by K. Weierstrass, un- published till 1894 (Werke, Vol. 1, p. 67), that the latter must have drawn the distinction as early as 1841. The concept of uniform convergence did not become common property till much later, chiefly through the lectures of Weierstrass. Other forms of the definition of uniform convergence. 3 rd form. 2f n (x) is said to be uniformly convergent in J f ^f ) in whatever way we may choose the sequence 10 (x n ) in the interval J\ the corresponding remainders invariably form a null sequence 11 . We can verify as follows that this definition is equivalent to the preceding: a) Suppose that the conditions of the 2 nd form of the definition are fulfilled. Then, given e, we can always determine N so that | r n (x) \ < 6 for every n > N and every x in /'; in particular I r n( x n) I < e * or ever y n> N; hence r n (x n )-+0. b) Suppose, conversely, that the conditions of the 3 rd form are ful- filled. Thus for every (o; n ) belonging to /' , r n (x n ) * . The conditions of the 2 nd form must then be satisfied also. In fact, if this were not the case, if a number N= -^(e) with the properties formulated there did not exist for every e > , this would imply that for some special e, say e = , no number N had these properties; above any number N, however large, there would be at least one other index n such that, for some suitable point x = x n in /', | r n (x n ) \ ^ e . Let n li be an index such that | r n (x n ) | ^ s . Above n l there would be another index 2 , such that | r n2 (x n ^ \ ^ e for a suitable corresponding point x n ^ and so on. We can choose (x n ) mj' so that the points x n , x n ^ . . . belong to (x n ), in which case 10 The sequence need not converge, but may occupy any position in./'. 11 Should each of the functions | r n (x) \ attain a maximum in J' t we may choose x n in particular so that | r n (x n ) \ = Max | r n (x) \ ; our definition thus takes the special form: Zf n (x) is said to be uniformly convergent in /' if the maxima Max | r n (x) \ in J' form a null sequence. If the function | r n (x) \ does not attain a maximum in _/', it has, however, a definite upper bound /i w . We may also formulate the definition in the general form: Form 3 a. 2f n (x) is said to be uniformly convergent in J' if p n -> 0. (Proof?) 46. Uniform convergence. 335 will certainly not form a null sequence, contrary to hypothesis. Our assump- tion that the conditions of the 2 nd form could not be fulfilled is inadmissible; the 3 rd form of the definition is completely equivalent to the 2 nd . In the previous forms of the definition, it was always the remainder of the series which we estimated, the series being already assumed to con- verge. By using portions of the series instead of infinite remainders (v. 81) the definition of uniform convergence may be stated so as to include that of convergence. We obtain the following definition : 4 th form. A series Ef n (x) is said to be uniformly convergent in the interval /' if, given s > 0, we can assign a number N = TV (e) depending only on e, and independent of x, such that for every n > N, every k^.1 and every x in /'. For if the conditions of this definition are satisfied, then it follows firstly (by 81) that f n (x) converges for each fixed x iny'. In the inequality, we may make k tend to QO, and we find that | r n (x) \ ^ e for each x in J'. Conversely, if | r n (x) | <g s for all n > N and all x in /', then for all these n, all k ^> 1, and all x in /', we have l/n+i (x) + +f n+k (x) | - | r n (x) - r n+k (x) I ^ 2 . This shows, however, that if the series Hf n (x) satisfies the conditions of the 4 th form, it also satisfies those of the 2 nd form, -and conversely. We may finally express this definition in the following form (cf. 8 la): 5 th form. A series Sf n (x) is said to be uniformly convergent in the interval J' if, when positive integers k ly k 2 , & 3 , . . . and points x^ x 2J # a , . . . of J' are chosen arbitrarily, the quantities [/nfl (*n) +/n +2 (*n) + +/nf/r n (*n) ] invariably form a null sequence 12 . Further Examples and Illustrations. 1. The student should examine afresh the behaviour of the series Zf n (x), with 192* a) in the interval 1 < x < 2, b) in the interval ^ x 5^ 1 (cf. the considerations on pp. 330 1). 2. For the series 1 + (x - 1) + (x* - x) + - + (x n - x n ~ l ) +. 12 By 51, we might even write C/^+i (*) + . . . +A w +* n (*n)] for the above, where the v n 's are any integers tending to + <. Exactly as in 81, we may speak of a sequence of portions, except that here we may substitute a different value of x in each portion. The statement we then obtain is: A series 2f n (x) is said to be uniformly convergent in J' if every sequence of portions of the series forms a null sequence. Similarly: A sequence of functions s n (x) is said to be uniformly conver- gent inj' if every difference-sequence is a null sequence. 336 Chapter XI. Series of variable terms. we have obviously s n (x) = x n . The scries accordingly converges in the inter val /: 1 < x < -f- 1 , in particular in the sub-interval /': < x < 1 . Here F(x) = for < x < I . = 1 for x = 1 . The convergence in this interval is not uniform. It is not so even in /": < x < 1 ; for here r H (x) = F(x) s n (x) = x n . We have only to choose in /" (hence in /') the sequence of points x =1- (n=l 2 ) n n \ 1 9 ) I 1 V 1 to have r n (x n ) I 1 ) * , so that the series cannot converge uni- \ n I e formly 13 . This may be made clear geometrically by examining the position of successive curves of approximation, as illustrated by the accompanying figure : For large values of n , the curve y ~ s n (x) remains, almost throughout the whole interval quite close to the #-axis, which represents the limiting curve. Just before the ordmate as = +l, it rises abruptly until it reaches its terminal point (1, 1). However large a value may be assumed for n, the curve y = s n (x) will never remain close to the limiting curve throughout the entire l * inter val y (or./'). 3. In the preceding example, we could almost expect a priori that the convergence would not be uniform, as F(x) itself has a "jump" of height 1 at the endpoint of the interval. The case was different with the example treated on p. 330. An example similar to the latter, but even more striking, is the following: Consider the series for which Fig. 6. nx (11=1,2,...). For a! = 0, we have s n (0) = , for every n\ for x =)= , the number e~~~* x * is positive and less than 1, so that (by 38,1) s n (x) *0. Our series is therefore convergent for every x and its sum is F(a;) = 0, i. e. the limiting curve coin- cides with the a; -axis. The convergence is not in the least uniform, however, if we consider an interval containing the origin. Thus, for x n = rr=- V" ' which certainly does not -> . The approximation curves have a similar For x n = f 1 g-J , we even have r n (x n ) -+ 1 . 14 In spite of this, it is easy to see that for every fixed x (in < x < 1) the values s n (x) diminish to as n increases, so that the abrupt rise to the height 1 occurs to the right of x, however near x may be taken to + 1, provid- ed only that n is chosen sufficiently large. 46. Uniform convergence. 337 appearance to those m Figs. 4 and 5, with this modification, that the height of the hump now increases indefinitely with n ; this is because 15 ~+ -r-OO. 4. We i mst emphasize particularly that uniform convergence does not require each of the functions f n (x) to be individually bounded. The series L_ 1 4- a; -f x 2 + , for instance, is uniformly convergent in < x <i -~- , with x ' the sum ., j^ ., since the remainders have the value x n \-x == t2n-l The first term of this series (as also the limiting function) is not bounded in the interval in question. (Cf., however, theorem 4. below.) With a view to calculation with uniformly convergent series, it is convenient to formulate the following theorems specially, although the proofs are so simple that we may leave them to the reader: Theorem 1. // the p series 2f n ^ (x), 2 f n2 (x), . . ., 2f np (x) are, simultaneously, uniformly convergent in the same interval J, (p is a definite whole number), the series 2 f n (x) for which is also uniformly convergent in that interval, if c 19 c a , . . ., c denote any constants. (I. e.: Uniformly convergent series may be multiplied by constant factors and then added term by term?) Theorem 2. // 2f n (x) is uniformly convergent in J, so is the series 2 g(x}f n (x), where g(x) denotes any function defined and bounded in the interval /. (I. e.: A uniformly convergent series may be multi- plied term by term by a bounded function.) Theorem 3. // not merely 2f n (x), but H\f n (x}\ is uniformly convergent in /, then so is the series 2g n (x)f n (x), provided that when m is suitably chosen, the functions g m ^\(x], g m +*(x), ..., are uniformly bounded in J, i. e. provided we can find an integer m > and a number G > such that \ g n (x) \ < G for every x in J and every n > m. (7. e.: A series which still converges uniformly when its terms are taken in absolute value may be multiplied term by term by any functions all but a finite number of which, at most, are uniformly bounded in J.) 15 The point for which x ~ is actually the maximum point of the curve v* y s n (x), as may be inferred from s' n (n n 9 x 2 ) e~~ * nx = . 338 Chapter XI. Series of variable terms. Theorem 4. If Sf n (x) converges uniformly in /, then for a suitable m the functions f m+l (x), f m+2 (x), . . . are uniformly bounded in J and con- verge uniformly to 0. Theorem 5. If the functions g n (x) converge uniformly to inj, so do the functions y n (x) g n (x\ where the functions y n (x) are any functions defined in J and with the possible exception of a finite number of them uniformly bounded in J. We may give as a model the proofs of Theorems 3 and 4: Proof of Theorem 3. By hypothesis, given e > 0, we can de- termine n Q > m so that for every n > and every x in J y For the same n's and #'s we then have |ft+i/.+i + I 5S I g.+i I |/.+i I + < G (!/ ,-i I + . . .) < e. This proves all that was required. Proof of Theorem 4. By hypothesis, there exists an m such that, for every n ^ m and every x in y, | r n (x) \ < \. Hence for n > m and every x in /, I/. (*) I = I r.-i (*) - r n (x) | ^ | r.-! | + | r n | < 1, which proves the first part of the theorem. If we now choose n Q > m so that for every n ^ n Q and every x in J, | r n (x) | < i e, (e being previously assigned) the second part follows in quite a similar way. v -** 47. Passage to the limit term by term. ^ * Whereas we saw on pp. 328 9 that the fundamental properties of the functions f n (x) do not in general hold for the function F (x) repre- sented by 2f n (x), we shall now show that, roughly speaking, this mil be the case when the series is uniformly convergent 16 . We first give the following simple theorem, which becomes particularly important in applications: 193. Theorem 1. // the series 2f n (x) is uniformly convergent in an interval and if its terms f n (x) are continuous at a point x n of this interval^ the function F (x) represented by the series is also continuous at this point 17 . 16 We may, however, mention at once that uniform convergence still only represents a sufficient condition in the following theorems and is not in general necessary. 17 If x is an endpoint of the interval y, only one-sided continuity can of course be asserted at X Q for F (x), but of course only the corresponding one-sided con- tinuity need be assumed at x for / n (x). 47. Passage to the limit term by term. 339 Proof. Given s > 0, we have (in accordance with 19, Def. 6b) to show that a number d = d (e) > exists such that | F (x) F (x ) | < e for every x with | z X Q \ < 8 in the interval. Now we may write F(x) - F(* )= s n (x) - sM + r n (x) - r n (x Q ). By the assumed fact of uniform convergence, we can choose n = m so large that, for every x in the interval, | r m (x) \ < . Then The integer m being thus determined, s m (x) is the sum of a fixed number of functions continuous ata? , and is therefore (by 19, Theorem 3) itself continuous at X Q . We can accordingly choose 6 so small that for every x in the interval for which )# X Q \ < (5, we have For the same x's we then have which establishes the continuity of F(x) at a; . Corollary. // 2f n (x] = F(s) is uniformly convergent in an interval, and if the functions f n (x\ are all continuous throughout the interval, then so is F(x). ' In connection with example 3 of 191,2, we have in the above a fresh proof of the continuity of the function represented by a power series in its in- terval of convergence. If we use the lim-defmition of continuity (v. 19, Def. 6) instead of the e- definition, the statement of the theorem may be put into the form: "m ( jy n (X)) n=0 In this form it appears as a special case of the following much more elaborate theorem: Theorem 2. We assume that the series F(x) = ^f n (x) is uni- 194. n=o formly convergent in the open interval 18 x . . . x l and that the limit, when x approaches X Q from the interior of the interval 19 , lim /(*) = * 18 X Q may be > or < #, . Whether the series remains convergent at a? , and indeed whether the functions f n (a;) are defined there at all, is immaterial for the present theorem. 19 We are therefore concerned here, as also in the two subsequent state- ments, with a one-sided limit. 340 Chapter XI. Series of variable terms. 00 exists. The series 21 a n ^ en converges and\imF(x), when x+x in n o the above manner, exists. Moreover , if we write 2a n = A , we have lim F(a) A, or, otherwise, n=0 (The latter form is expressed shortly by saying: In the case of uni- form convergence , we may proceed to the limit term by term.) Proof. Given e > 0, first choose w , (v. 4 th form of the defini- tion 191) so that for every n > x , every k ^> 1 and every x in our interval, Let us for the moment keep n and k fixed, and make x+x . By 19, Theorem la, it follows that And this is true for every n > n and every & ^> 1. Hence 2 a n is convergent. Let us denote the partial sums of this series by A n and its sum by A. It is easy to see now that F(x) >A. If, for a given e, n Q is determined so that, for every n > w , we not only have then, for a (fixed) m > n , |F(*)-4| = !(*.(*) - A J ~(A- ^+r m (x)\^\s m (x) - A m \ +-5- As a; *-a? involves 5 m (a?) >^4 m , we can determine 5 so that for every x belonging to the interval, such that < | x X Q \ < d . For these a; f s, we then also have \F(x)-A\<*, which proves all that we required. If (o; n ) is chosen arbitrarily in the interval of uniform convergence, it follows from and n, (a? n ) -* (v. 191, 3 rd form) that the sequences F(x n ) and s n (x n ) will in- variably exhibit the same behaviour as regards convergence or divergence, and that if they converge, the limits will coincide. We may contrast this with the case 47. Passage to the limit term by term of the series, already seen to be non-umformly convergent, whose partial sums are s n (x) -r~ir~ir~* If here we take x n , we have F (x n ) = 0, i. e. it is con- vergent with the limit 0, whereas s n (x n ) J, i. e. it also converges, but with the limit J. The two sequences do not have the same behaviour. Theorem 3. The series F(x) = Ef n (x) is assumed uniformly con- 195. vergent in the interval/, and all the functions f n (x) are supposed integrable over the closed sub-interval /': a ^ x ^ b, so that F (x) is also continuous in that sub-interval. Then F (x) is also integrable over J' and the integral of F (x) over the interval/' may then be obtained by term-by -term integration , i . e. b b b F(x)dx or f\Vf n (x)}d X = S [//( J L n =0 J 7i-0 '-' a a a (More precisely: The series on the right hand side is also convergent and has for its sum the required integral of F (x). Proof. Given s > 0, we determine m so large that for every n > m and every x in a ... b, f J Since s m (x) is the sum of a finite number of integrable functions, it is itself integrable over/'. By 19, theorem 11, we can therefore divide the interval J' into p parts t l9 i 2 , . . . , ij> such that, if ov denotes the oscil- lation of s m (x) in it, , we have < *=i Now the oscillation of r m (x) is certainly < 2 ?7, 1:~S > ky t ^ ie manner ' n which m was determined. Also the oscillation of the sum of two functions is never greater than the sum of the oscillations of the two functions. So for the same subdivision i l9 i 2 , . . . , i v of the interval a . . . b, we have i v < e, where v v denotes the oscillation of .F (x) in *" . Thus (again by 19, theorem 11) F(x) also is integrable over/'. Furthermore, as F = s n + r n> we have, for every n^m, b b b fF(x)dx- fs n (x)dx = fr n (x)dx <-J<c f a a a the latter by 19, theorem 21. Now s n (x) is the sum of a finite number of functions; applying 19, theorem 22, we therefore at once obtain b b f F(x)dx- E ff(x)d a ~~ a (061) 342 Chapter XI. Series of variable terms. b Tliis, however, implies the convergence of 2 J f v (x)dx and the iden- a tity of its sum with the corresponding integral of F(x). Matters are not so simple in the case of term-by-term differen- tiation. In 190, 7, we saw, for instance, that the series converges for every #, and so represents a function F(x) defined for every real x. The terras of this series are, without exception, continuous and dilfer- entiable functions. If we differentiate term by term, we obtain the series 00 2 cos n x , n = l which is divergent 20 for every x. - Even if a series converges uniformly for every x, as for instance the series (cf. Kxample 5, 191, 2), the position is no better, since on differentiating term by term we obtain V cos n x ~i n a series which diverges e. g. for x = Q. The theorem on term-by-term differentiation must accordingly be of a different stamp. It runs as follows: CO 196. Theorem 4. Given 21 a series E f n (x) whose terms are differen- n=0 liable in the interval /= a . . . b y (a < b); if the series 1 /;'(*). n=0 deduced from it by differentiating term by term, converges uniformly in /, then so does the given series, provided it converges at least at one point of J. Further t if F(x) and (p(x)are the functions represented by the, two series, F(x) is differentiable f and we have *(*)-?(*). In other words, with the given hypotheses, the series may be differ- entiated term by term. 20 The formulae established on p. 357 give, for every x ={= 2 k JT, 1 sin + ! jr- -f cos x + cos 2 x + + cos n x = 2 ' 21 As regards the convergence of the series, no assumption is made in the first instance. 47. Passage to the limit term by term. 343 Proof, a) Let c denote a point of / (existent by hypothesis) for which 2f n (c) converges. By the first mean value theorem of the differential calculus ( 19, theorem 8) 2 (/,(*)- (<)) = (*- c)' 2 //(), v=n+l r-n + l where f denotes a suitable point between x and c. Given e > 0, we can, by hypothesis, choose n so that for every n > n Q , every k ^> 1, and every x in J, 6 a ' Under the same conditions, we therefore have < e. n+fc 27 ( r=n H This shows that 2(f n (x) f n (c)), and hence Sf n {x) itself, is uniformly convergent in the whole interval / and accordingly represents a de- finite function F(x) in that interval. b) Now let X Q be a special point of / and write A(*o + A)-A.(*o) = gy(/t)> (,, = 0,1,2,...). These functions are defined for every h^O for which X Q -f- & belongs to /. As above, we may write n + ls n+fc 27 ^ (*) = 27 /;'(* + **) (o < * < i) r=fl y=+l and we find, as in a), that w 27 ^ n W n=0 converges uniformly for all these values of h. This series represents the function h By theorem 2, we may let h >0 term by term, and we conclude that F* ( x o) ex i sts > Wlt ^ ^(*o) = 270 ft, (*)) = n=0 A->0 n=0 This signifies that F*(x ) = q>(x ), as asserted. Examples and Remarks. 1. If a n (x xj* has the radius f>0 and if < ^ < r, the series (a? sfy)*"" 1 converges uniformly for every | x x \ < Q. By theorem 4, the given power series accordingly represents a function which is differentiable for every \x X Q \<Q. For any particular x, with \x # | < r, which we may choose to consider, we can determine p < r so that | x X \ <C g < r. The 344 Chapter XI. Series of variable terms. function represented by 2a n (x-~x ) n therefore remains differentiable at every point of the open interval | x x | < r. 2. The function represented by ^ - - is differentiable for every x and its derived function is - ^ . (Cf, Example 5, 191, 2.) n = l n 3. The condition of uniform convergence is certainly sufficient in all four theorems. But it remains questionable whether it is also necessary. a) In the case of the continuity -theorem 1 or its corollary, this is cert- ainly not so. The scries considered in 192, 2 and 4 have everywhere-con- tinuous terms and represent everywhere-continuous functions themselves. Yet their convergence was not uniform. The framing- of necessary and sufficient conditions is not exactly easy. S. ArzelA (Rendiconti Accad. Bologna, (1), Vol. 19, p. 85. 1883) was the first to do so in a satisfactory manner. A simplified proof of the main theorem enunciated by him will be found in G. Vivanti (Rendiconti del circ. matem. di Palermo, Vol. 30, p 83. 1910). In the case in which the functions / (x) are positive t it has been shown by V. Dim that uni- form convergence is also necessary for the continuity of F(x). Cf. Ex. 158. b) The fact that in theorem 195 on term-by-term integration uniform convergence is again not a necessary condition may also be verified by various CO examples. Taking the series fn ( x ) discussed on pp. 330-1, whose partial n=i sums are . nx and whose sum is F(a?) = 0, we see at once that v=l 00 Thus term-by-tcrm integration leads to the correct result. In the case of the i series 192,3, however, in which we also have J F(x) dx = 0, term-by-term o integration gives, on the contrary, In this case, therefore, term-by-term integration is not allowed. 48. Tests of uniform convergence. Now that we are acquainted with the meaning of the concept of uniform convergence, we shall naturally inquire how we can de- termine whether a given series does or does not converge uniformly in the whole or a part of its interval of convergence. However difficult it may be and we know it often is so to determine the mere convergence of a given series, the difficulties will of course be considerably enhanced when the question of uniform convergence 48. Tests of uniform convergence. 345 is approached. The lest which is the most important for applications, be- cause it is the easiest to handle, is the following: Weierstrass 9 test. // each of the functions f n (x) is defined and 197. bounded in the interval /, say throughout J and if the series 2y n (of positive terms) converges, the series 2f n (*t) converges uniformly in J. Proof. If the sequence (# n ) is chosen arbitrarily in /, we have By 81, 2, the right hand side *0 when n *oo; hence so does the left. By 191, 5 th form, 2 f n (x) is therefore uniformly conver- gent in /. Examples. 1. In the example 191, 3 we have already made use of the substance of Weierstrass* test. 2. The harmonic series Jj? , which converges for x >1, is uniformly convergent on the semi-axis #>l + <5, where d is any positive number. In fact, for such s's, 1 < 1 where 2y n converges. This proves the statement. The function represented by the harmonic series known as Riemann's ^-function and denoted by (a) is therefore certainly continuous for every 2a x> 1. 3. Differentiating the harmonic series term by term, we deduce the series This again is uniformly convergent m#^>l-|-<5>l. In fact, for every suffi- logn n 6 we then have 1 log n 1 <C 7T~ ciently large n, -<1 (by 38,4); for these n's and for every #> n Riemann's f -function is accordingly difFerentiable for every x > 1, and its derivative is represented by the series (*). 4. If 27 a n converges absolutely, the series 2 a n cos n x and 2 a n sin n x are uniformly convergent for every x, since e. g. | a n cos n x \ ^ a n = y w . These series accordingly define functions continuous everywhere. In spite of its great practical importance, Weierstrass* test will necessarily be applicable only to a restricted class of series, since it 22 In fact, if we consider a special x > 1, we can always assume 8 > chosen so that x > 1 + 8. 346 Chapter XI. Series of variable terms. requires in particular that the scries investigated should converge absolutely. When this is not the case, we have to make use of more delicate tests, which we construct by analogy with those of 43. The most powerful means for the purpose is again Abel's partial summation formula. On lines quite similar to those already followed, we first ob- tain from it the 00 198. Theorem. A series of the form 2 a n ( x }'^n( x } certainly converges n-^O uniformly in the interval /, if, in /, 00 1) 2jA v -(b p b v+1 ) is uniformly convergent (as a series) and v=0 2) (A n -6 n + 1 ) is uniformly convergent (as a sequence) 2 *. Here the functions A n = A n (x) denote the partial sums of 2 a n (x). Proof. As formerly we have merely to interpret the quant- ities a v , b v and A v as no longer numbers, but functions of a; we first have n+k n+fc 2*, \ = 2A r -(b r -b r ^) + (A n+t .b n+1l+1 -A n .b n+1 }. v^n-t-1 r=n+l Letting x and k vary in any manner with n, we have on the left z sequence of portions of the series 2 a v b v , and on the right the corresponding one relative to the series 2A v (b v & y+1 ), and a difference-sequence of the sequence G^n'^n+i)' Since by hypothesis the latter sequences always tend to (v. 191, 5 th form), it follows that so does the sequence on the left. This (again by 191, 5), proves the statement Exactly as in 43, the above theorem, which is still very general in character, leads to the following more special, but more easily man- ageable tests 24 : 1. Abel's test. 2 a v (x) b v (x} is uniformly convergent in /, if 2a v (x) converges uniformly in /, if further, for every fixed value of x, the numbers b n (x) form a real monotone sequence 25 and if, for 23 fl n , b n , A n are now always functions of x defined in the interval /; only for brevity we often leave the variable x unmentioned. For the notion of the uniform convergence of a sequence of functions cf. 190, 4. 24 For simplicity's sake, we name these criteria after the corresponding ones for constant terms. Cf. p. 315, footnote 8. 25 Cf. footnote to 184, I. 48. Tests of uniform convergence. 347 every n and every x in J, the functions b n (x) are less in absolute vahie than one and the same number 28 K. Proof. Let us denote by a n (x) the remainder corresponding to oo the partial sum A n (x); i.e. a v (x] = A n (x) + #(#) In the formula n=o of Abel's partial summation, we may (by the supplement 183) sub- stitute u for A r , and we obtain n\-k 2 <V&,= - 2 <* v=n M v=n + 1 it therefore again suffices to show that both 2a v (b v b v + l ) and (a n -& n + 1 ) converge uniformly in /. However, the# n (#/s, as remainders of a uniformly convergent series, tend uniformly to and the b v (xjs remain < K in absolute value for every x in /; it follows that (a n 'b n + 1 ) also converges uniformly to in /. On the other hand, if we con- sider the portions n\-k we can easily show that these tend uniformly to in /, thereby completing the proof of the uniform convergence in / of the series under discussion. In fact, if a v denotes the upper bound of a v (x) in /, -*() (v. form 3 a). Thus if f % is the largest of the numbers & n + l9 n+l \T n \< V 2\b v - 6, +1 | ^ vl&.-M - v=n+l involves the fact that T" n *0 uniformly in /. 00 2. Dirichlet's test. J a^ (a;)- 6 n (x) is uniformly convergent in /, n-O t'/ ^^ partial sums of the series 2 a n (x) are uniformly bounded 26 in f and if the functions b n (x) converge uniformly to in J, the conver- gence being monotone for every fixed x. Proof. The hypotheses and 192, 5 immediately involve the uniform convergence (again to 0) of (A n 'b n + :L ). If, further, K' denotes !ft The b n (x)'s form, for a fixed x } a sequence of numbers b Q (x), 6 (#), . . . ; for a fixed n, however, b n (x) is a function of x, defined in /. The above as- sumption may, then, be expressed as follows: All the sequences, for the various values of x, shall be uniformly bounded with regard to all these values of x\ in other words, each one is bounded and there is a number K which is simtdtaneously a bound above for them all. Or again: All the functions defined in / for the various values of n shall be uniformly bounded with regard to all these values of n\ i. e. each function is bounded, and a number J\ exists which simultaneously exceeds them all in absolute value. 348 Chapter XI. Series of variable terms. a number greater than all the |^4 n (a;)|'s for every x 9 we have n + k n+k 2j I b v b v+l | <* K -\b n + le+1 b n + l \ r=n+l In whatever way x and & may depend on n, the right hand side will tend to by the hypotheses, hence also the left. This proves the uni- form convergence in / of the series under consideration. The monotony of the convergence of b n (x) for fixed x has only been used in each of these tests to enable us to obtain convenient upper estimations of the portions 2\b y b v + l \. By slightly modifying the hypotheses with the same end in view, we obtain 3. Two tests of du Bois-Reymond and Dedekinrt. a) The series 2 a v (x) b v (x) is uniformly convergent in J, if both 2 a v and 2 \ b v & v+ i | converge uniformly in J and if, at the same time, the functions b n (x) are uniformly bounded in /. Proof. We use the transformation n+k n+k As the remainders cc y (x) now converge uniformly to 0, we have, for every v > m, say, and every x in /, | <*(#)! < 1. Hence for every n ;> m, n+k n\-k ^ 2\b,-b, +l \; r=n-fl the expression on the right even if x and k are made to depend on n, in any manner now tends to as n increases, hence so does the expression on the left. That # M -& nfl tends uniformly to in / follows, by 192, 5, from the fact that a n (x) does and that the 6 n (oj)'s are uniformly bounded in /. b) The series S a v (x) b v (x) is uniformly convergent in J if the series S\b v b v+1 \ converges uniformly in J y and the series 27 a v has uni- formly bounded partial sums, provided the functions b n (x) -> uniformly in J. Proof. From the hypotheses, it again follows at once that A n b n Hl converges uniformly (to 0) in /. Further, if K' once more denotes a number greater than all the |^ n ()|'s for every x, n+k '-Sl&r-W vn+l whence, on account of our present hypotheses, the uniform conver- gence in / of the series 2 A v (b v & y + 1 ) may at once be inferred g 48. Tests of uniform convergence. 349 Examples and Illustrations. 1. Tn applications, one or other of the two functions a n (x) and b n (x) 199. \vill often reduce to a constant, for every n\ it will usually be the former. Now a series of constant terms 2 a v must, if it converges, of couisc be re- garded as uniformly convergent in every interval; for, its terms being independent of #, so are its portions, and any upper estimation valid for the latter is valid ipso facto for every x. Similarly the partial sums of a series of constant terms 2a y , if bounded, must be accounted uniformly bounded m every interval^ 2. Let (a n ) be a sequence of numbers with -T n convergent, and let b n (x) = x n . The series 2a n x n is uniformly convergent in < x < 1 , for the conditions of AbeV test are fulfilled in this interval. In fact, 2a n , as re- marked in 1., is uniformly convergent; fuither, for every fixed x in the inter- val, (x n ) is monotone and | z n | < 1. By the theorem 194 on term-by-terin passage to the limit, we may therefore conclude that lim (2a n x") = (lim a n x n ), i. e. This gives a fresh proof of Abel's limit theorem 100. 3. The functions b n (x) =- - also form a sequence bounded uniformly in / (namely, again < 1), and monotone for every fixed x. Hence, as above, we deduce that " -*.. if 2 a n denotes a convergent series of constant terms. (Abel's limit theorem or Dinchlet series.) 4. Let a n (x) cos n x or =sinng, and &() = -, >*0. The series n n v 5 ! / N t / N vi cos nx \i sm M x i ^ t\\ a n (x).b n (x) = - or \] ---- _, (a > 0) , n=l n-1 n-1 "* then satisfy the conditions of DiricMet's test in every interval of the form. 27 5 ^ x ;S 2 TT 8, where 8 denotes a positive number < IT. In fact, by 185,5, the partial sums of 2a n (x) are uniformly bounded in the interval /we may take K --- - \ and b n (x) tends monotonely to 0, uniformly, because b n does not depend on a:. If (b n ) denotes any monotone null sequence, it follows for the same reason that ,2" b n cos n x and S b n sin n x are uniformly convergent in the same intervals (cf. 185, 5). All these series accordingly represent functions which are defined and continuous 28 for every 87 Or in intervals obtained from the above by displacement through an integral multiple of 2 jr. 28 Every fixed x ={= 2 k n may indeed be regarded as belonging to an inter- val of the above form, if 8 is suitably chosen (cf. p. 343, example 1, and p. 345, footnote). 12* (G51) 350 Chapter XI. Series of variable terms. x =j= 2 k IT. Whether the continuity subsists at the excluded points x = 2 k TT we cannot at once determine, not even in the case of the series 27 b n sin n x t although it certainly converges at these points (cf. 216, 4). 49. Fourier series. A. Euler's formulae. Among the fields to which we may apply the considerations developed in the preceding sections, one of the most important, and also one of the most interesting in itself, is provided by the theory of Fourier series, and more generally by that of trigonometrical series, into which we now pro- pose to enter 29 . By a trigonometrical series is meant any series of the form 1 2 a Q + 2 (a n cos n x + b n sin n x), w=i with constant 30 a n and b n . If such a series converges in an interval of the form c^x<c + 2Tr, it converges, in consequence of the periodicity of the trigonometrical functions, for every real #, and accordingly represents a function defined for all values of x and periodic with the period 2 TT. We have already come across trigonometrical series convergent everywhere, for instance, the series, occurring a few lines back, ^ ~ t ^ ., .fr-~' a 0; B ?r~^~' a 1; etc - We have never been in a position, so far, to determine the sum of any of these series for all values of x. It will appear very soon, however, that trigonometrical series are capable of representing the most curious types of functions such as one would not have ventured to call functions at all in Euler's time, as they may exhibit discontinuities and irregularities of the most complicated description, so that they seem rather to represent a patchwork of several functions than to form one individual function. 29 More or less detailed and extensive accounts of the theory are to be found in most of the larger text books on the differential calculus (in particular, that referred to on p. 2, by H. v. Mangoldt and K. Knopp, Vol. 3, 8 th ed., Part 8, 1944). For separate accounts, we may refer to H. Lebesgue, Lecons sur les series tngono- me'triques, Paris 1906, and to the particularly elementary Introduction to the theory of Fourier's series, by M. Backer, Annals of Math. (2), Vol. 7, pp. 81 152. 1906. A particularly detailed account of the theory is given by E. W. Hobson, The theory of functions of a real variable and the theory of Fourier series, Cambridge, 2 ud ed., Vol. 1, 1921, and Vol. 2, 1926. The comprehensive works of L. Tonelli, Sene trigonometnche, Bologna 1928, and A. Zygmund, Trigonometrical series, Warsaw 1935, are quite modern treatments; the little volume by W. Rogostnski, Fouriersche Reihen, Sammlung Goschen 1930, is particularly attractive and con- tains a wealth of matter. 80 It is only for reasons of convenience that a Q is written instead of a . 49. Fourier series. A. Euler's formulae. 351 Thus we shall see later on (v. 210 a) that e. g. :0 for z = ft:T,'(ft = 0, 1, 2, ...), but -x / = or x = n sin n x 2j \ _ (2 ft + !)*-!! =i I - o for 2 2 (ft + the function represented by this series thus has a graph of the following type: * Fig. 7. Similarly, we shall see (v. 209) that = for n = ft n, but ^i ^ j| \t-> '* ~r j n^O 2 w + X for 2 ft JT < :r < (2 ft -f- 1) ar, and = - ~ for thus the function represented by the series has a graph of the type: A - 2ji - .?r Fig. 8. In either case, the graph of the function consists of separated stretches (unclosed at either end) and of isolated points. However, the circumstance that simple trigonometrical series such as the above are capable of representing functions which are them- selves altogether discontinuous and "pieced together", is precisely what was chiefly responsible for the thorough revision to which the concept of function, and thence the whole foundation of analysis, came to be subjected at the beginning of the 19 th century. We shall see that trigonometrical series are capable of representing most of the so-called "arbitrary functions" 31 ; in this respect, they constitute a far more powerful instrument in higher analysis than power series. 81 Of course the concept of an "arbitrary function" is not sharply defined. The term usually denotes a function which cannot be assigned by means of a single closed, formula (i. e. one avoiding the use of limiting- processes) in terms 352 Chapter XI. Series of variable terms. We will mention only incidentally that the range of this instru- ment is by no means restricted to pure mathematics Quite the con- trary: such scries were first obtained in theoretical physics, in the course of investigations on periodic motion, i. e. chiefly in acoustics, optics, electrodynamics, and the theory of heat; Fourier, in his Thorie de la chaleur (1822) instituted the first more thorough study of certain trigonometrical series, although he did not discover any of the fundamental results of their theory. What functions can be represented by trigonometrical series and by what means can we obtain the representation of a given function, sup- posing this to be feasible? In order to lead up to a solution of this question, let us first assume that we have been able to represent a particular function f(x) by a trigonometrical series convergent everywhere: On account of the periodicity of the sine and cosine functions, f(x) is then necessarily periodic with the period 2n, and it is suffi- cient, therefore, to consider any interval of length 2n. We choose this interval, for all that follows, to be < x <j 2 n, where one of the end- points inay, moreover, be omitted. The function f(x) is then represented in this interval by a con- vergent series of continuous functions. We know that f(x) may none the less be discontinuous, although it also will be continuous if the series in question converges uniformly in the interval For the moment, we will assume this to be the case. With these hypotheses, we obtain a relationship between f(x) and the coefficients a n and b n which was conjectured by Euler: of the so-called elementary functions alone, i. e. in particular, it denotes a function which is apparently built up from separate portions of simple func- tions of this type, like the functions given as examples in the text, or the following, defined for every real x: for irrational x x for rational x , etc. Cf., however, the "arbitrary" function expressed by means of limiting processes on p 329, footnote. Not until it was found that even a perfectly "arbitrary" function such as these could be represented by a single (relatively simple) expression, as for instance by our trigonometrical series or by other limiting processes, did any necessity arise for regarding it as being actually one function, instead of a mere patchwork of several functions. 49. Fourier series. A. Euler's formulae. 853 Theorem 1. The series 200. -0-00+ J;(0 w cosn& + & w sinna;) 6 n=l is assumed uniformly convergent** in the interval 0<Ja?<l27r, with the sum f(x). Then for n = 0, 1, 2, ..., (Euler or Euler-Fourier formulae) 33 . Proof. As is known by elementary considerations, the following formulae 34 hold for every integral p and q (^> 0): a ) J cospx-cos qxdx o 2* b) / cos px- sin qxdx = o V ( = c) I sin px sin qxdx \ o [ =7i = for /> + g ===== JT for p = > = 2 rc f or /> = = o = for p =j= and p = for = Let us multiply the series for /"(#), which is uniformly convergent in 0<^x<^2n, by cospx; by 192,2 the uniformity of the conver gence is not destroyed, and after performing the multiplication we may accordingly (v. 195) integrate term by term from to 2n. We immediately obtain: .-t J f(x)cospxdx ' 1 f = V a o J cospxdx for p = 6 o = a / cospx'Cospxdx for ^; 32 In consequence of the periodicity of cos re and sin a;, it is then, ipso facto, uniformly convergent for every x. 88 This designation is a purely conventional one; historical remarks are given by H. Lebesgue, loc. cit., p. 23; A. Sachse, Versuch einer Geschichte der trigonometrischen Reihen, Inaug.-Diss., Gottingen 1879; P. du Bois-Reymond, in his answer to the last-named paper; as well as very extensively by H. Burk- hardt, Trigonometrische Reihen und Integrale bis etwa 1850 (Enzyklop. d. math. Wiss., Vol. II, 1, Parts 7 and 8, 191415). 84 We have only to transform the product of the two functions in the integrand into a sum in accordance with the known addition theorems, fe. g. cos p x cos q x = -jr- [cos (/> q) x + cos (p + q) xn , in order to be able to integrate straight away. 354 Chapter XI. Series of variable terms, i. e in either case jt j f(x)cospxdx; for the remaining terms give, on integration, the value 0. In the same way, multiplying the assumed expansion of f(x) by sinpx and then integrating, we at once deduce the second of Eider's formulae 2 Jt b p = I f(x)s'mpxdx. The value of this theorem is diminished by the number of assumptions required to carry out the proof. Also, it gives no indi- cation how to determine whether a given function can be expanded in a trigonometrical series at all, or, if it can, what the values of the coefficients will be. However, the theorem suggests the following mode of procedure: Let f(x) be an arbitrary function defined in the interval <[ x <J 2 n, and integrable in Riemann's sense in the interval. In that case the integrals in Euler's formulae certainly have a meaning, by 19, theorem 22, and give definite values for a n and b n . We therefore note that these numbers, exist, on the single hypothesis that f(x) is integrable. The numbers -^- a Q , a lf a^, . . . and b 13 b 9 ... thus defined by Euler's formulae will be called the Fourier constants or Fourier coefficients of the function f(x). The series -?r &(\ + J? f # cos n x 4- b sin n x) 2 ^ n ~^ n ^ " ' may now be written down, although this implies nothing as regards its possible convergence. This series will be called (without reference to its behaviour or to the value of its sum, if existent) the Fourier aeries generated by, or belonging to, f(x) 9 and this is expressed symbolically by /()~4- This formula accordingly implies no more than that certain constants a n> b n > have been deduced from f(x) (assumed only to be integrable) by means of Euler's formulae, and that then the above series has been written down 35 . 3i The symbol "/-v/" has of course no connection here with the symbol introduced in 4O, Definition 5, for "asymptotically proportional". There is no fear of confusion. 49. Fourier series. A. Euler's formulae. 355 From theorem 1. and the manner in which this series was derived, we have, it is true, some justification for the hope that the series may converge and have f(x) for its sum. Unfortunately, this is not the case in general. (Examples will be met with very shortly.) On the contrary, the series may not converge in the whole interval, nor even at any single point; and if it does so, the sum is not necessarily f(x). It is impossible to say off-hand when the one or the other case may occur; it is this circumstance which prevents the theory of Fourier series from being entirely a simple subject, but which, on the other hand, renders it extraordinarily fascinating; for here entirely new problems arise, and we are faced with a funda- mental property of functions which appears to be essentially new in character: the property of producing a Fourier series whose sum is equal to the function itself. The next task is then to elucidate the connection between this new property and the old ones, viz. con- tinuity, monotony, differentiability, mtegrability, and so on. More con- cretely stated, the problems which arise are therefore as follows: 1. Is the Fourier series of a given (integrable) function f(x) con- vergent for some or all values of x in < x <T 2 n ? 2. // it converges, does the Fourier series of f(x) have for its sum the value of the generating function ? 3. If the Fourier series converges at all points of the interval u^ x ^P> is the convergence uniform in this interval? As it is conceivable that a trigonometrical expansion of f(x) might be obtained by other means than that of Euler's formulae, we may also raise the further question at once: 4. 7s it possible for a function which is capable of expansion in a trigonometrical series to possess several such expansions, in par- ticular, can it possess another trigonometrical expansion besides the possible Fourier expansion provided by Euler's formulae? It is not very easy to find answers to all these questions; in fact no complete answer to any of them is known at the present day. It would take us too far to treat all four questions in accordance with modern knowledge. We shall turn our attention chiefly to the first two; the third we shall touch on only incidentally, and we shall leave the last almost entirely out of account 36 . 36 It should be noted, however, that the fourth question is answered under extremely general hypotheses by the fact that two trigonometrical series which converge in ^ x ^ 2 TT cannot represent the same function in that interval without being entirely identical. And if/(#), the function represented, is integrable over . . . 2 TT, its Fourier coefficients are equal to the coefficients of the trigono- metrical expansion; cf. G. Cantor (J. f. d. reine u. angew. Math., Vol. 72, p. 139. 1870) and P. du Boit-Reymond (Munch. Abh., Vol. 12, Section I, p. 117. 1870). 356 Chapter XI. Series of variable terms. With the designations introduced above, the content of Theorem 1 may be expressed as follows: Theorem la. If a trigonometrical series converges uniformly in 5^ x 5^ 2 TT (t. e. for all #), it is the Fourier series of the function repre- sented by it, and this function 37 admits of no other representation by a trigono- metrical series converging uniformly in ^ x < 2 TT. The fact that the Fourier scries of an integrable function does not neces- sarily converge will be seen further on; that even when it does converge, it need not have / (<v) for its sum, is obvious from the fact that two different functions f (x) and / 2 (x) may very well have identically the same Fourier constants ; in fact two integrable functions have the same integral (and therefore the same Fourier con- stants; i. e. the same Fourier series), if they coincide, for instance, for all rational values of x, \\ithout coinciding everywhere (v. 19, theorem 18). The fact that in an interval of convergence the series need not converge uniformly is shown by the example already used above; for the series J converges everywhere (v. 185, f>), and if the convergence were uniform, say in the interval 8 ^ x ^ 8, 8 0, it would have to represent a continuous function in that interval, by 193. This is not the case, however, as we mentioned before on p. 351 and will prove later on p. 375. These few remarks suffice to show that the questions formulated above are not of a simple nature. In answering them, we shall follow the line adopted by G. Lejeune-Dirichlet, who took the first notable step towards a solution of the above questions, in his paper Sur la convergence des series trigonometriques 38 . B. Dirichlet's integral. We proceed to attack the first of the proposed problems, namely, the question of convergence: If the Fourier series - a + E (a n cos n x + b n sin n x) generated by 2i a given integrable function /(#), i. e. with coefficients given in terms of f(x) by Euler's formulae, is to converge at the point x = X Q , its partial sums 1 " + b v sin v X Q ) must tend to a limit when n -> + oo . It is often possible to determine whether or no this is the case, by expressing s n (x ) in the form of a definite integral as follows: 87 This function is then (by 193, Corollary) everywhere continuous. 88 Journ. f. d. reine u. angew. Math., Vol. 4, p. 157. 1829. 49. Fourier series. B. Dinchlet's integral. 357 For i>^l, the function a v cos v X Q + b v sin v X Q is represented by 39 2,-t 2 ~ I f(f)cosvtdt\cosvx + j Thus We now take the important step of replacing the sum of the (n -f- 1) terms in brackets by a single closed expression. We have indeed 40 for every z 4= %kn, for every a and all positive integral w's, cos (a + ,?) + ( os ( + 2 2) H ----- 1- cos ( + /w-) 201. sin f + ^ m + 1 -~-J sin f + -5-) 2 sin 25 / , -- rr Z\ sin HI ~cos f -j" m 4~ 1 ~) 39 In order to distinguish the parameter of integration from the fixed point a; , we henceforth denote the former by t. 40 Proof. If the expression on the left is denoted by C mt we have + sin *-) + sin (a + 2T+T -|- r=2sinm-^-.cos fa-f w + I ~J . Moreover the above formula continues to hold for z = 2 A JT , provided we attrib- ute to the ratio on the right hand side the limiting value for z i. e. the value wcosct. 358 Chapter XI. Series of variable terms. from which many analogous formulae may be deduced as particular cases 41 . Taking a = 0, z = t X Q> m = n, we obtain -5- + cos (t X G ) -\ -{- cos n(i x ) ~~~ O~ I ^ - - Accordingly 42 , sin(2n+l)-^5L (a) ' N ' " /A Finally, we may transform this expression somewhat. The function f(x) need only be defined in the interval 0<^x<,27i and integrable over this interval. The latter property remains unaltered if we merely modify the value of f(2jr) (cf. 19, theorem 17). We will equate it to /*(()) and define f(x) further, for every x such that i)jz, (A> = 1, 2, by: 41 For subsequent use, we mention the following: JT -.--{-a substituted for a gives: g / g sin m sin f a -j- m -f 1 - sin(a-f jr) + sin(4-2*)H {-sin(r ' - x sin T sin w cosm+ 1 ~ 2. of = gives: cosz-j- cos2^H .+ cos mz = ; z sin sin m - - - sin m -}- 1 -~- 3. a = -5- gives : sin ^ 4- sin 2 s -f - -f- sin w* = ; t . Z sin 4. = 2a;, a = y 05, give: / , \ , / , o \ , / , o 7 % sin wo; -cos (y 4- ma;) cos (y -f- x) -f cos (y 4- 3 a:) -| (- cos (y -f- 2 m 1 -a;) = r v -\ sin x 5.* = 2x, u = ~+y-x, give: S in( y + ^) + sin( y+ 3 a! ) + ... + sin( y + 2^^. g ) = SinWa - s ^ + ma;) . 42 For /rrajy, as we observed once before, we should attribute to the sine-ratio the limiting value for / *a; , here (2w-f 1) . 49. Fourier series. B. Dirichlet's integral. 359 Our function f(x) is now defined for all real values of x and we have arranged for it to be periodic with period 2n. Now for any function (p (x) periodic with period 2n, we have (by 19, theorem 19), what- ever the values of c and c' may be, / (p(f)dt = / cp(t)dt = / (p(c 9 -f- *)^* anc * / <P W<^ = / <p(t)dt. c a a-f-2?c As the integrand in (a) is now a function of this type, we have 8 /? ^ J sin T If we split up this integral into the pans relative to the intervals to 7i and n to 2n, substituting t for t in the second, the latter becomes - 2w t r sin(2n+l)-=- * sin TT - 2 i. e. by the above remark with regard to J <p(t)dt a sin(2n+l) ~ o . - sin i and we accordingly obtain o 2 Substituting 2 / for t, we are ultimately led to the formula This is Dirichlet's integral* 3 , by which the partial sums of the Fourier series generated by f(x) may be expressed. We may therefore state, as our first important result, the theorem: Theorem 2. In order that the Fourier series generated by a junc- tion f(x), integrable (hence bounded) and periodic with period 2n, may 43 We designate as. Dinchlet's integrals all integrals of either of the two forms a a sinkt , / or 360 Chapter XL Series of variable terms. converge at a point x Q9 it is necessary and sufficient that Dirichlefs integral 1 ( n J sin/ , should tend to a (finite) limit as n * + oo . TiWs fo'wutf xs then the sum of the Fourier series at the point # . Let us denote this sum by S(X Q ). The second question (p. 355), concerning the sum of the Fourier series, when convergent, may be included in our present considerations and our result may be put in a form still more advantageous in the sequel, by expressing the quantity S(X Q ) in the form of a Dirichlet integral also. As sin (2 *+!)- -H- + cos t + cos 2 1 -\ ----- (- cos nt = we have or, effecting the same transformations as before with the general integral, o Multiplying this equation 44 by 5 (# ), we finally obtain, by subtraction from 202, - rf t Our preceding theorem may now be expressed as follows: 2O3 Theorem2a. /w order that the Fourier series generated by a function f(x), integrable and periodic with period 2ir, should converge to the sum s (x ) at the point X Q , it is necessary and sufficient that, as n -> + 00 , Dirichlet' $ integral 44 This equation may also be obtained from 202, by substituting f (x) ~ 1 ; this gives a = 2 and, for every n 7-: 1, a n - b n - 0, i. e. s n (X Q ) = 1 for every n and every x . 49. Fourier series. 13. Dinchlet's integral. 361 should tend to 0, where for brevity we have put Although this theorem by no means solves questions 1 and 2 in such a manner that the answer in given concrete cases lies ready to hand, yet it furnishes an entirely new method of attack for their solution. Indeed the same may be said with regard to the third of the questions proposed on p. 355, for theorem 2 a may at once be modified to the following: Theorem 3. On the assumption that the partial sums s n (x) converge to s (x) at every point of the interval a 5^ x 5^ ft, they will converge uniformly to this limit in the interval, if, and only if, the integral, depending on x 9 J sin t u tends uniformly to as n - > -f - oo in a 5^ x fg ft, that is to say if, given e > 0, we car assign N = N (e) so that this integral is less than in absolute value for every n > N and every x in a rg x fg ft. Before we make use of theorem 2 to construct immediate tests of convergence for Fourier series, we proceed first to transform and simplify this theorem in various ways. For this purpose, we begin by proving the following theorems, which apparently lead us rather off the track, but also claim considerable interest in themselves. Theorem 4. Iff(x) is integrable over ... 2 77, and if (a n ) and (b n ) <n are its Fourier constants, then 2 (a n 2 -f- b n 2 ) converges. 71-1 Proof. The integral 27t n J [/(O 2 ( a <- cos v t 'I- ** sin vi\ 2 dt i' 1 is ^ 0, as its integrand is never negative. On the other hand, it is '2x 2n 27i f [/(O] 2 dt2[a, J/ (*) cos v id t] 2 2 [b v J/(f) sin v td t] + J [27 (a v cos v t + b v sin v t)] 2 d t o dt - 2 where each summation is extended from v = 1 to v = n. Since this expression is non-negative, we have 362 Chapter XI. Series ot variable terms. Thus the partial sums of the series (of positive terms) in question are bounded and the series is convergent, as asserted. The above contains in particular Theorem 5. The Fourier constants (aj and (6 n ) of an integrable function form a null sequence. From this, we may deduce quite simply the further Theorem 6. // y>(t) is integrable in the interval a<^t^b, then b A n / tp()cosn tit -+ 0, /* = / ty(f) sin nt dt -> 0. a Proof. If a and b both belong to one and the same interval of the form 2 k n <I t <^ 2 (k + 1) n , we define f(t) = y; (f) in a <[ t <L b and f(i) = at the remaining points of the first-named interval, for every other real t, f(t) is defined so as to be periodic with the period 2n. Then b 2t A n = f y (t) cos n tdt = / f(t) cos ntdt = na n a o and similarly B n = n b n , where a n and & M denote the Fourier constants of the function f(f). By theorem 5, A n and 5 W therefore 0. If a and 6 do not fulfil the above condition, we can split up the interval a < t < 6 into a finite number of portions, each of which satisfies the condition. A n and B n then appear as the sum of a (fixed) finite number of terms, each of which tends to as n *oo. Hence A n and B n do the same 45 . This important theorem will enable us to simplify the problem of the convergence of Dirichlet's integral 46 . Supposing 6 chosen arbitrarily with < d < -J , the function (/ ., o) |[n ~ 45 This important theorem appears intuitively plausible if we imagine the curve y \p (t) cos nt to be drawn for large values of n: We isolate a small interval a . . . ft in which y (t) has an almost negligible oscillation (is practically con- stant) and proceed to choose n so large that the number of oscillations of cosn* is fairly large in the interval; in that case, the arc of the curve y = y>(/)cosw/ corresponding to .../? will enclose positive and negative areas in approximately equal numbers and of approximately the same size, so that the integral is almost 0. 46 Of course theorem 6 may be proved quite directly, without first proving theorem 4. The latter is, however, an equally important theorem in the theory, even though, as it happens, we shall not need it again in the sequel. 49. Fourier series. B. Dirichlet's integral. 368 is integrable in 6^t^^. Hence, for fixed d, n ~2 c) / t/> (0 sin (2 n + 1) t - d t -* , The Dirichlet integral of theorem 2 a will therefore tend to as limit as w->oo, if, and only if for a fixed, but in itself arbitrary, value of d > the new integral 2^ o tends to as n increases. Now the latter integral only involves the values of f(x Q 2t) in 0^<<5, i. e. of f(x) in# 2 d<^x<*x Q + 2d. Since d > may be assumed arbitrarily small, this remarkable result contains at the same time the following Theorem 7. (Ricinann's theorem. 47 ) The behaviour of the 204. Fourier series of f(x) at the point X Q depends only on the values of f(x) in the neighbourhood of X Q . This neighbourhood may be as- sumed as small as we please In order to illustrate this peculiar theorem, we may mention the following consequence of it: Consider all possible functions f(x) (inte- grable in . . . 2 n) which coincide at a point X Q of the interval . . . 2 n and in some neighbourhood of this point, however small, possibly varying with the particular function. Then the Fourier series of all these functions however much they may differ outside the neigh- bourhood in question must, at X Q itself, either all converge or all diverge, and in the former case they have the same sum S(X Q ) (which may or may not be equal to f(x^)). After inserting these remarks, we proceed to re-formulate the criterion obtained above, which we may henceforth substitute for theorem 2: Theorem 8. The necessary and sufficient condition for the Fourier series of f(x) to converge at X Q to the sum s(# ), is that for an ar- bitrarily chosen positive 6 < -5- , Dirichlet's integral 3 _;_ /o i i \ A dt sin/ should tend to as n increases 48 . 47 liber die Darstellbarkeit einer Funktion durch cine trigonometrische Reihe, Hab.-Schrift, Gottingen 1854 (Werke, 2 nd ed. p. 227). 4 * As regards uniformity of convergence, we can assert nothing straight away, since we are ignorant as to whether the integral (c) above considered, which tends to as n increases, for every fixed a? , will do so uniformly for every 2 of a specified interval on the a;- axis. Actually this is the case, but we do not propose to enter into the question further. 364 Chapter Xf. Series of variable terms. There is no difficulty in showing that the denominator sin t in the last integrand may be replaced by t. In fact the difference be- tween the original integral and the one so obtained, i. e. the integral s automatically tends to as n increases, by theorem 6, because -T-T - is continuous and bounded 49 , and hence intcgrable, in < t^ d. Thus we may finally state: 205* Theorem 9. The necessary and sufficient condition for the Fourier series of a function f(x), periodic with the period 2n and integrable over Q...271, to converge to S(X Q ) at the point X Q , is that for an arbitrarily chosen positive d ( < -g- J , the sequence of the values of the integral 2 f , . v sin (2 n -f- 1) t ,^ - J <p (t ; X ) T- 2 dt o forms a null sequence. Here <p (i\ x () ) has the same meaning as in theorem 2 a. In another form, the condition is that, given e > 0, we can assign d < ? and N > 0, so that 50 for every n > N, 6 ^L f ft' \ sin (2 n + 1) t C. Conditions of convergence. Our preliminary investigations have prospered so far that the first two questions of p. 355 may now be attacked directly. By the above, these are completely reduced to the following problem: Given a function <p(t), integrable in Q<^t<*d, what further conditions must this function fulfil in order that the integrals 51 " In fact. -T : - ~ : -- = -\ ; in the interval, and thus itself 1 sin/ t /.sin / 1 h-" tends to as l-~*0. 60 The student should make it quite clear to himself that the second for- mulation is actually equivalent to the first, although d need only be determined after the value of e has been chosen. 61 For t = 0, we attribute to in the integrand the value A. 49. Fourier series. C. Conditions of convergence. 365 should tend to a limit as k increases, and what, in that case, is the value of this limit? 52 Since in this integral, & has a fixed but arbitrarily small value, the answer to this question depends only cf. Riemann's theorem 7 on the behaviour of q>(i) immediately to the right of 0, say in an interval of the form < t < <Jj (^ (5). We may accordingly inquire also: What properties must cp(f) possess immediately to the right of 0, in order that the limit in question may exist? A large number of sufficient conditions for this have been found, of which we shall only explain two, the great generality of which renders them sufficient for most purposes. The first of these was established by Dirichlet in the above-named paper (v. p. 356) and was the first exact condition of convergence in the theory of Fourier series, in winch Dirichlet's work is altogether fundamental. The second is due to U. Dini and was discovered in 1880. 1. Dirichlet' 's rule. // cp (t) is monotone to the right of 0, 206. i. e. in an interval of the form < t < d ( ^ <5) then the limit in question exists, and we have 6 ^ lim J h == lirn^ ^ j y w ^ o where y> denotes the (right hand) limiting value lim (p(), which cer- tainly exists with the assumptions made 53 . Proof. 1) In the first place, lim The existence of this limit, i. e. the convergence of the improper in- tegral, follows simply from the fact that, given e > 0, and any two o values #' and x" both > , we have (by 19, theorem 26) x" x" sin t , , hence x" u i 52 There is no simplification in observing that it would suffice tor k to tend to -f oo through odd integral values. 63 In fact, as <p (t) is integrablo, it is certainly bounded, and by hypothesis it is monotone in <C * <C <V Furthermore q? need not =97(0). 366 Chapter XI. Series of variable terras. Now, as we saw on p. 360, equation (b), the integrals f sin (2 n + A/ , *n=J Stai ** _j (2 n + 1) t 6 for n = 0, 1, 2, . . . , are all = ~ . Therefore we also have t w ~> ~ On the other hand, the numbers o (cf. the developments on p. 364) form a null sequence, by theoiem 6. Accordingly we also have n . _ V . , Psin (2 n -f- 1) * TT Since, however (v. 19, theorem 25), this implies that the above-named limit has the value ^-. 2) By 1), a constant K' exists such that ; 'o for every a?^0, and therefore a constant K(=2K') exists such that b a for every a, b such that ^ a <^ &. 3) Suppose given > and choose a positive <5' <i ^ , so that Writing 2 /A sin* t we then have f k J k ' tending to as k > + oo, by theorem 6, and we 49. Fourier series. C. Conditions of convergence. 3G7 can accordingly choose k 9 so large that | / fc / fc ' | < -g- for every k > A'. Further, T / 2 f A =-J sin ft * , . 2 sin ft o o For the second of these two quantities, we have * sin* . -,, J "ir^ and we may accordingly choose & > &' so large that for every A > A . For / fc ", the first of the two quantities on the right of (d), we use the second mean value theorem of the integral calculus 19, theorem 27), which gives, for a suitable non-negative d" <^ <5', The latter integral = -^-dt and therefore remains < K in absol- ^(5" ute value, by 2). Accordingly \ T"\ <1 . S .K <^ e I J x I = ^ 3~A' A < 3 ' Combining the three results of this paragraph, by means of A = (A-A') + A"-f/r> we see that, given e !> 0, we can choose k so that, for every k > ^ , Thereby the statement is completely established. 2. Dtni's rule. If lim cp (t) = q> exists, and if for every positive T <C d, the integrals d J l9 ' ( V* >o1 ^ T remain less than a fixed positive number, then lim/ fe exi ^ fr-V-f-oo More shortly: If the integral I ^-^ ^-^dt t which is improper at 0, has a meaning. 368 Chapter XI. Series ot variable terms. Proof. When r decreases to 0, the above integral increases mono- tonely but remains bounded; it therefore tends to a definite limit as z 0, which we denote for brevity by Given e > 0, we may choose a positive d' < 6 so small that J] (ft I Vo dt< ^' Writing, as in the previous proof, the difference (/ ft / k ') tends to 0, by theorem 6, and we may choose k' so large that | J k / k '| < -|- for every & > A'. Further, as we saw before, with a suitable choice of & > &' we also have 8 J for every k > & - Finedly, i. e. when d' is suitably chosen, 1 7 fc " | also remains < -q- . Thus, pre- cisely as before, we conclude that, for every k > & , which proves the validity of Dini's rule. We may easily deduce from it the two following conditions. 3. MpscMtz's rule. // two positive numbers A and a exist, such that 55 fn < / ^ <5, then J k +<p . Proof. 65 The "L^c^'te-condition", | 9> that lim rp (i) = cp Q exists. - n K ^'^ as ^-*0, itselt implies 49. Fourier series. C. Conditions of convergence. 369 so that for every positive t < d the former integral remains less than a fixed number and in consequence of Dints rule /,, ><p > as required. 4 tii ru ie. // q>' (0) exists** and therefore lim <p (t) = <p == 9? (0) exists, then A* 9V <->+o Proof. The existence of implies the boundedness of this ratio in an interval of the form 0<^<(5j, i.e. the fulfilment of a Lipschitz- condition with a = l. Hence / fc *<?V as asserted. The following corollary to these conditions is immediately ob- tained Corollary. // cp (f) can be split up into the sum of two or more functions, each of which satisfies the conditions of one of the four rules above, then lim cp (t) = (p again exists, and the Dirichlet integrals J k t->+o of the function (p(t) tend to <p Q . The above rules may at once be transferred to the Fourier series of an integrable function f(x), which we assume from the first to be given in <^ x < 2 n and to be extended to all other real values of x by the equation In order thai the Fourier series generated by f (x) should converge to a sum s(# ) at the point X Q , the integrals ,5 T 2 f f . N sin(2tt-f I)/ .. /H = -jJ v(*;*o) S ~ dt o must, by theorem 9 (&05), form a null sequence, where, as before, v (<; *) = \ [/K + 2 + /(* - 2 f)] - s K) . This form of the criterion shows, over and above Riemanris theorem 2O4, that neither the behaviour of f(x) immediately to the right of a? , nor that immediately to the left of # , have in themselves any influence whatever on the behaviour of the Fourier series of f(x) at X Q . What is important is that the behaviour of f(x) to the right of X should stand in a certain relation to that on the left of x , namely, such that the function ) ~ il/K + 20 +ffo> - 2 ft] - S(X Q ) 56 It suffices that cp' (0) should exist as the rv*ht hand differential coefficient (v. 19, Def. 10), as in fact the possible values of 9 (t) for t ~^* do not come into account. 370 Chapter XI. Series ot variable terms. should possess the necessary and sufficient properties 57 for the existence of the limit of Dirichlet's integrals J k (206) relative to q)(t). It is not known what these properties are. The four conditions given above for the convergence of Dirichlet's integrals furnish us, however, with the same number of sufficient conditions for the con- vergence, at a special point X Q , of the Fourier series of a function f(x). Each of these conditions requires, in the first instance, that the function should tend to a limit <p . A common assumption for all the rules which we are about to set up is accordingly the following: The limit (8) Km i [f (* + 2*) -/(*,,-- 2 <)] - must exist. The value of this limit, by theorem 2, will then also be the sum of the Fourier scries of f(x) at X Q> if the latter converges. This convergence is ensured if the function o) = l/(*o + 2 f) + f(* 9 ~ 2 <)] - s (* ), considered as a function of t> fulfils one of the four conditions given above. At the same time, the value <p Q in those conditions must, by theorem 2 a, be 0. We accordingly assume that the two following conditions are satisfied: 207. 1 st assumption. The function f(x) is defined and integrable (hence bounded] in the interval ^ x < 2 n and its definition is extended to all real values of x by means of the relation *1, 2,... 2 nJ assumption. The limit ln[f(x + 2t) + f(x -2()], where X Q denotes an arbitrary real number, but is kept fixed throughout, exists 58 , and its value is denoted by s (# ), so that the function 9 (0 = 9 (': *o) = | [/(* + 2 + /(* - 2 I)] - i (*o) has a right hand limit lim 9 (f) = 0. With these joint assumptions, we have the following four criteria for the convergence of the Fourier series off(x) at the point x : 87 Define e. g. / (x) as entirely arbitrary to the right of x (but integrable in an interval of the form x < x < x + 8) and, in x - 8 < x < x 0> let f(x) = 1 ~ /( 2 *o x) say. The Fourier series off(x) at x is convergent with the sum 2 > (Proof, for instance, by means of Dirichlet's rule 208, 1 below.) 68 The two-sided limit then necessarily also exists. 49. Fourier series. C. Conditions of convergence. 371 1. IMrichlct's rule. // <p(t) is monotone in an interval of the 208. form < t < <5j, the Fourier series of f(x) converges at X Q and its sum 59 is equal to S(X Q ). 2. Dint's rule. // for a fixed (otherwise arbitrary) positive num- ber d the integrals p dt remain less than a fixed number for every r such that < T < d, the Fourier series of f(x) converges at x and its sum is S(X Q ). 3. Lipschitz's rule. The same is true, if instead of requiring that the integrals should be bounded, we stipulate that two positive numbers A and a should exist, such that, for every t such that <t < d, 4 th rule. The same is true, if instead of the Lipschitz- condition we require that <p(t) should possess a right hand differential coeffi- cient at 0. The application of these rules is made considerably easier by the following corollaries: Corollary 1. The function f(x) also fulfils the assumptions 1 and 2 and its Fourier series converges at X Q to the sum s(# ), if f(x) can be split up into the sum of two or any fixed number of functions, each of which satisfies these two joint assumptions (for a suitable s) and in some neighbourhood of x fulfils the conditions of one of the above rules. Corollary 2. Similarly, it suffices to stipulate in place of assump- tion 2 that each of the two (one-sided) limits lim f(x + 2 f) = f(x + 0) and lim f(x - 2 1) = f(x - 0) t->+0 f-v+0 should exist, and that the two functions ^=/t*o + 2 ')-/'(*o + ) and <p, (t) = f(x - 2 t) - f(x Q - 0) should each, individually, satisfy the conditions of one of the four rules. The Fourier series of f(x) is then convergent at X Q and has the sum * (*o) = ![/(*(> + 0) + /X*o -<>)] One or two special cases, which, however, are of particular im- portance in applications, may be mentioned in the following further corollaries: 69 In case it converges at X QJ the Fourier series of a function f(x) satis- fying the assumptions 207 accordingly has the sum /*(#) if, and only if, the limit s(a? ), whose existence is stipulated in the second assumption, =f(x ). Similarly in the case of the following rules. 872 Chapter XJ. Series of variable terms. Corollary 3. If f(x) satisfies the first assumption and is monotone both to the right and to the left of x , the limits mentioned in the preceding corollary exist, and the Fourier series of f(x) converges at X Q to the sum s(z ) = y [f(x Q + 0) -f- f(x 0)]. Hence, still more particularly: Corollary 4. The Fourier series of a function f(x) which satisfies the first assumption will converge at the point x and its sum will be the value f(x) of the function at that point, if f(x) is continuous at X Q and monotone on either side of X Q . Corollary 5. If f(x) satisfies the first assumption, and the two limits f(x Q 0) exist; if, further, both the (one-sided) limits Um o-o and exist; then the Fourier series of f(x) will converge at X Q and will have the sum s (X Q ) = ~ [f(x Q + 0) + f(x - 0)]. Corollary 6. The Fourier series of a function f(x) which satisfies the first assumption will converge, and will have as its sum the value of the function, at any point X Q at which f(x) is differentiate. 50. Applications of the theory of Fourier series. As we see from the rules of convergence developed above, extremely general classes of functions are represented by their Fourier series. This we propose to illustrate by a number of examples. The function f(x) to be expanded must always be given in the interval <^ x < 2 n and must possess the period 2 n: f(x 2 n) = f(x). The corresponding Fourier series is then, in general, obtained in the form In particular cases, the sine- or cosine-terms may be absent In fact, if f(x) is an even function, (the graph of f(x) is symmetrical with respect to the straight lines x = k n, (k = 0, 1, 2, . . .), and therefore 2n n 2n n - b n = / f(x) sin n x d x = f + / = , n as is evident if we replace x by 2 TT x in the second of these two partial integrals. The Fourier series of f(x) thus reduces to a pure 50. Applications of the theory of Fourier series. 373 cosine-series. If, on the other hand, f(x) is an odd function, (the graph of f(x) is symmetrical with respect to the points x = kn, k = 0, 1, 2, . . .), and therefore 271 n . a n = / f(x) cos n x dx = 0, o as is equally evident. Thus here the Fourier series of f(x) reduces to a pure sine series. There are accordingly three different ways in which an arbitrary given function F(x), which is defined and integrablc in a <^ x <^ b, may be prepared for the generation of a Fourier series. l t method. If b a ^> 2 n, a portion of length 2 n is cut out of the interval (a, 6), say cc^,x<a-}-2n, and the origin is transferred to the point a; we thus obtain a function f(x) defined in <^ x < 2 71: It is then defined for the whole #-axis 60 by means of the condition of periodicity f (x -^- 2 TT) = /(#). If 6 # < 2 ?r, define /(#) to be con- stant = F (b) in b^x<.a+27T and proceed as before 61 . 2 nd method. Precisely as above, define a function f (x) in 5^ x ^ TT (not 2 77) by means of /? (#), put / (#) / (2 77 #) in TT fg # <I 2 TT, and then define f(x) for all further #'s by the condition of periodicity. 3 rd method. Define/(#) as above for < x < TT, put/(0) = /(TT) = 0, but put f(x)~ /(2 TT ^) in ?r < jc < 2 TT; then again define /(#) for all further x's by the condition of periodicity. The three functions which aie obtained by these methods from a given function F(#), and which are now suitable for the generation of a Fourier series, we shall distinguish as f^(x) t f%(x}, f 9 (x)- Whereas /*j (#) will certainly give a pure cosine scries and f% (x) a pure sine- series, f (x) will lead, as a rule, to a Fourier series of the general form (unless, in fact, f (x) is itself already an odd or an even function). Since our rules of convergence enable us to recognize the con- vergence only at points X Q for which lim ~ [f(x + 2t) + f(x - 2 t)] ^ exists, it will be advisable to modify our functions further at the 60 If b a > 2 TT, a portion of the curve y = F (x) is left out of the repre- sentation altogether. If we wish to avoid this, we need only alter the unit of measure- ment on the x-axis so that the interval of definition of F (x) has the length 2 TT; i. e. we substitute a + - ^ - x for x. 77 01 Or else give the interval of definition of F (x) the exact length 2 TT by modi- fying the unit of measurement on the *-axis. 13 (G51) 374 Chapter XI. Series of variable terms, junctions 2 k TT by writing /(0)=/(2A7r)--:lim x -> I " whenever this limit exists. (This is certainly the case for / 3 (x), and provides the condition / 3 (0) = / 3 (2 k 77) = 0.) If this limit does not exist, the functional value f(2kir) does not come into account, as with our resources we cannot discover whether the Fourier series con- verges there or not. For corresponding reasons we have already put /a 00 = ^ above. We now go on to concrete examples. 209. 1. Example. F(x) ss a =f 0. Here fi(x] E= /g (a;) EE a, while we have to put for x = and x = n> a < x < n, a n < x < 2 7i. Dirichlet's conditions are evidently fulfilled at every point (inclusive of the junctions), for each of the three functions. The expansions obtained must accordingly converge everywhere and must represent the functions themselves. For f^(x] and f%(x), however, they are trivial, as they reduce to the constant term a Q = a. For f s (x) f however, we obtain: 2 JT * 2n = \f 3 (x)sm nxdx = \s\nnxdx ~ \smnxdx = \ smnxdx t n i. e. for even values of n, for odd values of n. The expansion accordingly is f / x 4 a r . , sin 3 x , sin 5 a; 5 or 3t . ~T in sin 3 a? . sin 5 a? at and at JT, -- in This establishes the second of the examples given on p. 351, and provides the sum of this curious series, of whose convergence we were 50. Applications of the theory of Fourier series. 375 already aware (v. 185,5) 62 . For x = ~, , ^, we obtain special series, with the first of which we are familiar m an entirely different connection (v. 122): 1,11, n 4- ^ - 1 _L J L_ _|_ . . . = * 13 r 17 ~ ' 3 ' _ - 11 JL - -4 13 ~ 2' 2. Example. F(x) ---- ax, (a 4= 0). Here o; in 0<#<27r, an at and at 2 rc, ax in <J a; <^ rc, a (2 7i B) in ?r <I # ax in < x < rrc, at and at TT, a2jr in n 2 rc, gives: After an easy calculation, the expansion of ( \ ' -L. s ' u * x i sin Hac , sin4ap , I T*- ftn in < at and at 210. n x which establishes the first of the examples of p. 351. Similarly, the expansion of f 9 (x) gives sin 3 x sin 4 a? , T" . x . (b) sin oc sin 2 ' + in at rr, a? ft in ft < 05 <[ 2 rr < Or, more shortly, _f*in -ir- at TT. < TT, 62 This and the following examples are already found, for the most part, in Euler's writings. Many others have been given by Fourier ; Legendre, Cauchy, Frullani, Dirichlet and others. They are collected together, in a convenient form for refer- ence, in H. Burkhardt, Trigonometnsche Reihen und Integrate bis etwa 1850, Enzyklopadie d. math. Wiss., Vol. II A, pp. 902920. 376 Chapter XI. Series of variable terms. The function f 2 (#), however, provides the expansion: cos a: , cos 3 x , cos 5 x , _ | 8 -Tz- "I 53 1 H3 h inn S The first of these expansions gives for x = the known series for ^; the third, for x -= 0, gives the series, also previously known to us (137), 1 + V + 5^ + 7 + ' ' ' = 8"' from which we may immediately deduce the relation i+i+i+i+...=v previously established (136, 156 and 189) in an entirely different way 83 . On comparing the two results, we obtain the remarkable fact that in < x ^ TT the function x is capable of the two Fourier expansions sin 3 x . and _ 4 rcos_x , cos 3y , cos 5 y _. . "1 ~~ lr L"! 7 " "^ 32 ' 6 ' J ' With a view to penetrating still further into the significance of these results, it is well to sketch the graphs of the function / (x) and a few of the corresponding curves of approximation. This we must leave to the reader, and we shall only draw attention to the following phenomenon: The convergence of the series 210 c is uniform for all #'s; not so that of the scries 210 a and b, since their sums are discontinuous, the 63 A fifth proof, quite different again, is as follows: The expansion 123 is uniformly convergent in f x 5 1, by the stipulations made in 123, together with 199, 2. Putting x sin t t we see that the expansion is uniformly convergent in ^- t f* ^ and may therefore be integrated term-by- term over that interval. Now this is shewn by a recurrence process, or by writing cos t = z and using Example 117 b. Hence at once 7r a = 1 _ "8 w== o(2w~+ l) a * This method was essentially given by Euler. (Cf. the note referred to in the foot- note 38 to 156.) 50. Applications of the theory of Fourier series. 377 first at 0, the second at TT. In the former case, the approximation curves lie close to the zigzag line representing the limiting curve along the whole of its length, whilst in (a) and (b) the corresponding state of affairs does not and cannot occur (cf. 216, 4). 3. Example. F (x) = cos a x (a arbitrary 64 , but 4= 0, 1, 2, . . .). a) We first form the function / 2 (x), and accordingly define f cos ax in 0^#lS7r | cos a (2 TT x) in TT ^ x < 2 TT; thus / 2 (#) is a function continuous everywhere, which by Dirichlefs rule will also generate a Fourier series continuous everywhere, which represents the function, and is necessarily a pure cosine-series. Here we have n JT TT a n 2 J cos a x cos w jc c/ x J [cos (a + w) ^ + cos (a w) A;] d x\ hence, as a was assumed not to be an integer, _ f i \ n 2 a sma TT 7Td n (^ l; g'lT^a Therefore the function / 2 (x) in ^ Jt: ^ 2 TT, or in other words the function cos OLX in TT ^ ^? ^ + TT, is represented by the series : For ^ ^^ TT, we obtain from this the expansion 117, previously deduced from entirely different sources: cos (X.TT 1 . 2 a . 2 a . 77 ---- = 7T COt a 7T = - + -= -- r= + -^ - ^, -(- * sin a TT a a 2 I 2 ' a 2 2 2 ' We thus enter the sphere of the developments of 24. Of course the other series expansions there deduced may also be obtained directly from our new source. Thus 212 gives for x _ TT_ ___!__ 2j* __ , _ 2 a __ 2 a __ __ ^ sin a TT ~ a a 2 - 1 2 ~*~ a 2 ~- 2 2 a 2 ~- 3 2 "^ Subtracting the cotangent expansion obtained just before, we further obtain 1 cos arr _ aw _ 4a 4a 4a ^ ~sirTa ~ ~" w mn ~2" "" "" a 2 ^T 2 "~ a 2 - 3 2 ~ a 2 ~^~5 2 "" " * " and so on. b) If we now similarly construct an odd function / 3 (#) from F (x) = cos a x, we have J cos a x in < # < TT, / 3 (#) = J at and at TT, [ cos a (2 TT je) in TT < x < 2 TT. 84 Because otherwise the cosine-expansion would become trivial. 378 Chapter XI. Series of variable terms. Here a may also assume integral values without reducing the result to a trivial one. The coefficients b n are obtained from integrals whose value is easily worked out, and they lead to the following expansions, valid in <x <TT: 213. a) for a 4= 0, 1, 2, . . . T .- -, tin * + -.-inSa + g, , sin 5 * j8) for a = i /> = integer \ [12 :rp sin * + 32 ~^ sin 3 * + ' ' ']> if /> is cospx = 4 r 2 . ^ 4 . . , .- . ., 2 sin 2 x + -rr sin 4 x + , it p is odd. p* 4 2 p 2 ' ' ' r From all the above series, innumerable numerical series may be deduced by taking particular values of x and a. 4. The treatment of F (x) sin a x leads to quite similar expansions. 5. If the function F (x) = log (2 sin ^ J is arranged for the genera- tion of a pure cosine series, we obtain the expansion, valid in < x < TT, 214. cos x + ^. + ^^ +... = - log (2 sin J) . It has, however, to be shewn by a special investigation that the result holds in spite of the fact that the function is unbounded in the neigh- bourhood of the points and 2 ?r, and therefore is not (properly) intc- grable. (Cf. 55, V below, where this will follow quite simply in another way.) 6. Example. F (x) = e yx + ~ ax , a 4= 0, is to be expanded in a cosine series. We have therefore to take f / \ t F (*) in ^ * ^ * J*W~~\F(2ir-x) in TT ^ x < 2 IT. After working out the extremely easy integrals giving the coefficients a nt we obtain JT go* +- g-qv _ _1 a , a ^ , which is valid in - TT 5=1 x ^ + TT. If we substitute e. g. x = TT and write t for 2 a TT for simplicity, we are led, after a few simple transformations, to the relation, valid for every t 4= IT _I _ 1 _ Al ^ 2 Z . 1- 50. Applications of the theory of Fourier series. 379 i. e. to an "expansion in partial fractions" of this remarkable function; its expansion in power series we can at once deduce from 24, 4, where the function -~ - t -JT? was considered, for our function reduces to the latter by multiplying by t 2 and adding 1. Various remarks. The very fact that trigonometrical series are capable of representing extremely general types of functions renders the question as to the limits of this capacity doubly interesting. As was already remarked, necessary and sufficient conditions for a function to be representable by its Fourier series are not known. On the con- trary, we find ourselves obliged to consider this as a fundamental property of functions, new of its kind, for all attempts to build it up directly by means of the other fun- damental properties (continuity, differentiability, mtegrability, etc.) have so far failed. We must deny ourselves the satisfaction of supporting this statement in all details by working out relevant examples, but we should nevertheless like to put forward a few of the facts in this connection. 1. One of the conjectures which will naturally be made at first sight is that all continuous functions are representable by their Fourier series. This is not the case, as du Bois-Reymond was the first to show by an example (Gott. Nachr. 1873, p. 571) . 2. On the other hand, to assume the function differentia ble as well as con- tinuous is more than is necessary, as is shown by Weierstrass* 66 example of a uni- formly convergent trigonometrical series, viz. 00 / 3 \ a n cos(b n irx) (0 < a < 1, b a positive integer, ab > 1 4- n), n=l \ ^ / which accordingly is the Fourier series of its sum (v. 200, 1 a), but which represents a function that is continuous but nowhere differentiable. 65 We now have simpler examples than that mentioned above. E. g. L. Fejer has given a very clear and beautiful example (J. f. d. reine u. angew. Math., Vol. 137, p. 1. 1909). 06 Abhandlungen zur Funktionenlehre, Werke, Vol. 2, p. 223. (First published 1875.) 380 Chapter XI. Series of variable terms. 3. Whether continuous functions exist whose Fourier series are everywhere divergent is not at present known. 4. A specially remarkable phenomenon is that known as Gibbs* phenomenon 67 , which was first discovered (by J. W. Gibbs) in connection with the series 2 10 a: The curves of approximation y =^ s n (x) overshoot the mark, so to speak, in the neighbourhood of * = 0. More precisely, let us denote by ( n the abscissa of the greatest maximum 6a of y = s n (x) between and IT and let y n be the corresponding ordmate. Then n -> ; but -r\ n does not -*- -, as we should expect, but tends to a value g equal to (1 17808 . . .). Thus it appears that the limiting configuration to which the curves y = s n (x) approximate contains, besides the graph of the function 210 a (p. 351, fig. 7), a stretch of the >--axis, between the ordinates g, whose 2 length exceeds the "jump" of the function by nearly -. In fig. 9, the th approxi- mation curve is drawn for n = 9 m the interval . . . ?r, and for n = 44 the initial portion is given. 51. Products with variable terms. Given a product of the form //a +/.(*)). = i whose terms are functions of x, we shall define (in complete analogy with the theory of series) as an interval of convergence of the product, an interval / at every point of which all the functions f n (x) are defined and the product itself is convergent. Thus e. g. the products /(: - 5). //(- + ). //(' + <- *).(. are convergent for every real x t and the same is true of any product of the form 77(1 + a n x), if 2 } a n is either absolutely convergent (v. 127, theorem 7) or a con- ditionally convergent series for which 27 a n 2 converges absolutely (127, theorem 9). For every x in f> the product then has a specific value and therefore defines a determinate function F(x) in /. We again say: the product represents the function F (x) in /, or: F (x) is expanded in the given product in /. The main question is as before: how far do the fundamental pro- perties (of continuity, differentiability, etc.) belonging to the terms f n (x) still hold for the function F (x) represented by the product? Here again the 67 J. W. Gibbs, Nature, Vol. 59 (London 1898-99), p. 606. Cf. also T. II. Gronwall, Ober die G&fasche Erscheinung, Math. Annalen, Vol. 72, p. 228, 1912. 68 The maxima in the interval occur at x = * ., ~- ., - * . . . , the 2 77 4 IT first being the greatest maximum. The minima occur at x = - f , . . . . n n 51. Products with variable terms. 381 answer will be that this is the ease in the widest measure, as long as the products considered are uniformly convergent. What the definition of uniform convergence of a product is to be is almost obvious if we refer to the corresponding definition for series, since in either case we are essentially concerned with sequences of functions (cf. 190, 4). However, we shall set down the definition corresponding to the 4 th form (191, 4) for series: Definition . The product II (1 +/(*)) is said to be uniformly 17 convergent in an interval /, //, given e > 0, a single number N = N (z) depending only on e, not on x, can be chosen so that for every n ~> N, every k > 1 and every x in /. It is not difficult to show that with this definition as basis the theorems of 47 hold substantially for infinite products 71 . We will, however, leave the details to the student, while we prove a few theorems which are less far-reaching, but which will amply suffice for all our applications, and which have the advantage of providing us at the same time with criteria for the uniformity of the convergence of a product. We first have Theorem 1. The product //(I + f n (x)) converges uniformly in /218. and represents a continuous function in that interval, if the functions f n (x) are all continuous in / and the series 2 \ f n (x) \ converges uniformly in /. Proof. If 2 1 f n (x) | converges in /, so does the product // ( 1 (- /(#)), by 127, theorem 7; indeed, it converges absolutely. Let F (x) denote the function it represents. Let us choose m so large that l/m+l (*) I + I/H 2 (*) I + + Ifm+K (*) | < 1 for every x in J and every k^l; this is possible, by hypothesis. Consider the product // (!+/(*)), n=m+l 69 The symbol o in this section again holds only with the same restrictions as in 4648; cf. p. 327, footnote 1. 70 This definition includes that of convergence. If the latter be assumed, we may speak of the "remainder" r n (x) (1 +/ n+ i (#))(! H f n +* 0*0) and define uniform convergence as follows: //(I +/ n (x)) is said to converge uniformly in/, if for every (x n ) in/, however chosen, r n (x) -> 1. Writing //(I +/(*)) = P m (x) and 77 (1 +/(*)) = F w (*), we may i/=l i> = rofl quite easily deduce e. g. the continuity of F (x) at x from that of the functions f v (x) there, by means of the relation F (x) - F (*) - P m (*) F m (*) - P m (*) F m (*) = [P m (x) - P m (*)] F m (x) + [F m (*) - F m (*)] P m (*). 13 (051) 382 Chapter XI. Series of variable terms. and denote its partial products by p n (x), n > m. Let F m (x) be the function represented by this product. We have (cf. 190, 4) Fm, ~ Pm+l + (Pm+2 ~~ Pm-\ l) + + (Pn Pn-l) + = Pm+l + Pm\ 1 */w+2 + Pmj 2 */w+3 + + Pn-l '/ + i. e. F m (x) is also expressible by an infinite series as is indeed evident from 30. Now (by 192, theorem 3) this series converges uniformly in J. In fact, for every n > m, we have and !/(*) I n*=M-\-l is uniformly convergent in /, by hypothesis. Accordingly the sequence of its partial sums, i. e. the sequence of functions p n (x), tends uniformly CO to F m in/, so that the product // (1 +/ n (x)) is seen to converge uniformly n -m+l in /, and this property is not affected when we prefix the first m factors. By 193, F m (x) is necessarily continuous in /, since the terms of the series which represents it are all continuous in that interval. The same is then true of the function F (x) = (1 + /i (*)) . . . (1 + f m (*)) F m (x), q. e. d. A similar proof holds for Theorem 2. // the functions f n (x) are all differentiable in J and if not only 2\f n (x) |, but \f n ' (x) \ converges uniformly in /, then F (x) is also differentiable in /. Moreover its differential coefficient at every point of J where F (x) 4= is given by 72 Proof. The proof may be put in a form analogous to that of the previous theorem; however, in order to make other methods of attack familiar, we will conduct the proof by means of the logarithmic function, as follows. Let us choose m so large that 72 If g (x) is difTerentiable at a special point x and g (x) 4= there, the ratio 8 - - is called the logarithmic differential coefficient ofg (x), because it is , log | g (x) |. For (x) = gi (x) - a (x) . gk (x), we have, as is well known, SLM _ Si' W , */(*> , , *(*> g(x) g, (x) "*" g,(x) "*--*- & ffi ' provided that the functions A (x) are all differentiable at the point x in question. 51. Products with variable terms. 383 for every x in /, so that, in particular, for every n > m, I /(*)!<! By 127, theorem 8, the series J? log (!+/(*)) fi=m + l is then absolutely convergent in /. The series obtained from it by differen- tiating term by term, __ !+/(*)' is indeed also uniformly (and absolutely) convergent in J. For since l/n (*) I < \ for ever y n > m y | 1 + f n (x) | > \ and therefore < 2, so that the uniform convergence of the last series follows from that of Z\f n ' (x) |. Accordingly (by 196) if, as before, we put //a +/(*)) = *",, n=m + l i. e. ^ log (1 +/(*)) = log F w (*). n m i-1 Since finally and the last factor on the right has been seen to be differentiate in J 9 F (x) itself is different! able in /. If, further, F (x) 4= 0, the last relation leads at once to the required result, by the rule of differentiation men- tioned in the preceding footnote. Applications. 1. The product 219, F m (*)- II (l-S) ( w>0 > n=mH X ' x 1 is uniformly convergent in every bounded interval, since, with f n (x) = , is evidently a uniformly convergent series in that interval. The product accordingly defines a function F m (x) continuous everywhere, which, in particular, is never zero in | * | < m -f 1. This function is also differentiate, for S \f n ' (x) | = 2 | x \ S- 1 384 Chapter XT. Series of variable terms. is uniformly convergent in every bounded interval. Hence for | x \ < m F m ^(x) _ 1 , f 2*_ *(*) * + n JZ+i*-* By 117, this however implies *'<*>- ff cot.. J J7 ** - G '<*> *v,(*) - w " w *-i- ^i. _ ,,-i - G TO (*)' where G m (#) denotes the function sin TT x interpreting this expression as equal, for x = 0, 1, . . . , ih w, to its limit (obvi- ously existent and -J- 0) as jc tends to these values. (The corresponding conven- tion is made for the middle term in the relation immediately preced ing.) If however two functions F (x) and G (x) have their logarithmic derivatives equal in an interval, in \vhich the two functions never vanish, it follows that they can only differ by a constant factor (4= 0). Hence, in | x \ < m -f- 1, sin TT x = c x // (l - 2 where c is a suitable constant. To determine its value, we need only divide the last relation by x and let x -> 0. The left hand side then -> ?r, while the right hand side -> c, because the product is continuous at x 0. Accordingly c TT and we have, first for | x \ < m + 1, but hence, as m was arbitrary, for all x t sin T x = 7i x . M---1 This product, and those discussed below m 2 and J:, as well as the remarkable product 257, 9, and many other fundamental expansions in products, are due to Eider. 2. For cos IT x we now find, without further calculation, sin 2 7T x " X COS7I* = n -- = 3. The sine-product for special values of x leads to important numerical product expansions. E. g. for A; == ^> /(2>-i)(2>i + " g j. i | As -> 1, we may clearly omit the brackets, and we accordingly write 2 ft 77 = 2 ' 2 ' 4 ' 4 ' 6 1 6 -! 8 ' 8 2~~ l-3-3-6-6-7-7 ; 9... (Wales' Product). Since it follows from this that /2\2 /4\ 2 / 2k 246 2& 1 - i" 3 '5 2~v- rv*~" V7r 78 Arithmetica infinitorum, Oxford 1656. (Cf. pp. 218 9, footnote 1.) Exercises on Chapter XI. 385 we obtain at the same tune the remarkable asymptotic relation 2. 4. 6.. .2* 2* for the ratio of the middle coefficient in the binomial expansion of the (2 w) th power to the sum 2 2n of all the coefficients of this expansion or for the co- efficient of x n in the expansion of . yl x 04. The sequence of functions (v. 128 4) cannot be immediately replaced by a product of the form 77(1 -f /*,(#)) as 77(1 -h J diverges for a; =J= 0- However, this divergence is of such a kind that By 1S8, 2 and 42, 3, this implies that = <r (*) tends, as n *-QO, to a specific limit, finite and 4" 0; the latter, of course, only if x -\- 0, 1, 2, . . . . Accordingly is a definite number for every x 4= 0, 1, 2, .... The function of x so de- fined is called the G a tn ma- function (T-junction). It was introduced into analysis by Euler (see above) and, next to the elementary functions, is one of the most important in analysis. Further investigation of its properties lies out- side the scope of this book. (Cf., however, pp. 439 440 and p. 630.) Exercises on Chapter XI. I. Arbitrary series of variable terms. 154* Let (nx) denote the difference between nx and the integer nearest to x, or the value 4-77, if nx lies exactly in the middle of the interval between two consecutive integers. The series ~^- * s uniformly convergent for all aj's. The function represented by it, however, is discontinuous for x ^~- ^ 2 q (p, q integers), while it is continuous for all other rational values of x and for all irrational values of x. 155. ff a n ->0, ^ / sin n x \* 2 *(- .IT) converges uniformly for all x's Does this remain true for a n ~ 1 ? 336 Exercises on Chapter XL 156. The products c) 77-l + sin-, d) // l +(_!)" sin converge uniformly in every bounded interval. 27l 157* The series whose partial sums have the values s n (x) = - converges for every x. Is this convergence uniform in every interval? Draw the curves of approximation. 158. A series 2 f n (x) of continuous positive functions certainly converges uniformly if it represents a continuous function F(x). (Cf. p. 344, Rem. 3.) 159. Does J>] ^= --- gr converge uniformly in every interval? Is the function it represents continuous? 160. In the proof of 111, a situation of the following kind occurred: An expression of the form F (n) = (n) + a, (n) + . + fc (n) + . . . + a fn (n) is considered, in which, for every fixed k t the term a^ (n) tcnJs to a limit a^ as n increases. At the same time, the number of terms increases, p n *> QO . May we infer that lira F(ri) = 2," fcf n-><x> /i=J provided the series on the right converges? Show that this is certainly per- missible if, for every h and every n t | a k (n) \ remains < y fc and 2," y k converges. Formulate the corresponding theorem for infinite products. (Cf. Exercise 15, where such term-by term passages to the limit were not allowed. > 161. The two series x 9 x 4 x 6 x> x 9 Q are both convergent for < x < 1 and have the same sum - - log 2 for x - -f- 1 . Li What is their behaviour when x * 1 0? Examine the two series, con- vergent for x >! 1 , 12112 for a;->4-H-0. 162. The series J? ^ -^ converges in < x <J 1 What is its sum? Is its convergence uniform? Kxercises on Chapter XI. 387 163. Show that, for x -> 1 + , 164. Show that, for #->! 0, a ) 2 ~ I rrr^r -" 4~ lo 2 * n=l * * M n ~\ Vf n-i w * n a C) (l-*)'Jt/ (- 1 ) i a:a"*""T' n=i 165. The series whose partial sums have the values s n (x) may not be integrated term by term over an interval with endpoint 0. Draw the curves of approximation. II. Fourier scries. 166. May we deduce from the series 210 a y by integration term by term: a ) ^-^~-~ fi=i '2 _ y _ _ In which intervals are these relations valid? (Cf. 297.) 167. In the same way, deduce from 210 c the relations * cos (2 n - - What would be the results of further integrations? In which intervals are these expansions valid? 168. From 209, 210, and the relations in the two preceding exercises, deduce the following further expansions and determine their exact intervals of validity: . cos 3 x , cos 5 x , n a) cos* -- 3 + g -- + ...= _, cos 3 x , cos 5 x b) v . sin 3 a; , sin 5 x , nx/n* x*\ C) n-- 84 +-^ -- + ... = _(___), etc. 169. From 215, deduce further expansions by substituting n x for x or by differentiating term by term. Is the latter operation allowed? What are the new series so obtained? 388 Chapter XII. Series ot complex terms. 170. What are the sine-series and the cosine-series for e ax ? What is the complete Founer expansion of e sma; ? Show that the latter is of the form a H- 6j sin x - a 1 cos 2 x 6 3 sin 3 x -\- a 4 cos 4 x -f- b. sin 5 X h 4- - 2 where a v and b v are positive. 171 If a; and y are positive and <C^r, ~- if /> ^ * * fl sinwscosny -2- Determine the values of the integrals If n Jsin ,r , , f sin a; , ax and I ax. x J x (The former = 1 '37498..., the latter = 1-8519 ) 173. For every x and every n, sin 2 # smx-i ^ r-- f sin a; J T-"- o where the bound on the right hand side cannot be diminished (cf. the preceding exercise). (Further exercises on special Fourier series will be given in the next chapter.) Chapter XII. Series of complex terms. 52. Complex numbers and sequences. After we have discussed in detail, as in Chapter I, the modes of formation of all the concepts essential for building up the system of real numbers, no new difficulties are raised by the introduction of further types of numbers Since the (ordinary) complex numbers and their algebra are known to the reader, we may accordingly be content with briefly mentioning one or two main points here. 1- I* was sho wn m 4 that the system of real numbers is in- capable of any further extension, and is, moreover, the only system of symbols satisfying the conditions which we laid down for a number system. Yet the system of complex numbers is a system of bymbols to which the name of number system is applied. This apparent contra- 52. Complex numbers and sequences. 389 diction is easily removed. For our definition of the number concept was in a certain sense an arbitrary one, as we emphasized on p. 12, footnote 16: A series of properties which appealed to us essential in the case of rational numbers was raised to the rank of characteristic pro- perties of numbers in general, and the result justified our doing this, in so far as we were able actually to construct a system in all essen- tials, a single one, which possessed all these properties. If we desire to attribute to other systems the character of a system of numbers, we must therefore of necessity diminish the list of char- acteristic properties which we set up in 4, 1 4. The question arises which of these properties may be dispensed with first of all; i. e. which of them may be missing from a system of symbols without its becoming impossible to legard the latter as a number system. 2 Among the properties 4 of a system of symbols, the first with which we may dispense, without fear of the system losing the character of a number system entirely, are the laws of order and monotony. These are based, by 4, 1, on the fact that of two different numbers of the system, the one can always be called less than the other, and the latter greater than the former. If we drop this distinction and in 4 replace both the symbols < and ;> by 4=> ^ appears that the modified conditions 4 are satisfied by another more general system of symbols, namely the system of ordinary complex numbers, but thai no other system substantially different from the latter can satisfy them 3. Accordingly, the system of (ordinary) complex numbers is a system of symbols which, as is known, may be assumed to be of the form x + yi, where x and y are real numbers, and i is a symbol whose manipulation is regulated by the single condition t 9 = 1, for which the fundamental laws of arithmetic 2 remain valid without ex- ception, provided the symbols < and > are suitably replaced throughout by =4=. In short: Except for the last-named restriction, we may work formally with complex numbers exactly as with real numbers. 4. In a known manner (cf. p. 8), complex numbers may be brought into (1,1) correspondence with the points of a plane and may thus be represented by these: with the complex number x --J- yi we associate the point (x, y) of an ay-plane. Every calculation may then be interpreted geometrically. Instead of representing the number x-\-yi by the point (x, y), it is often more convenient to represent it by a directed line (vector) coincident in magnitude and direction with the line from (0, 0) to (x, y). 5. Complex numbers will be denoted in the sequel by a single letter: z, f, a, b, . . .; and unless the contrary is expressly mentioned or follows without ambiguity from the context, such letters will in- variably denote complex numbers. 390 Chapter XII. Series of complex terms. 6. By the absolute value (or modulus) | z \ of the complex numbei x+yi, is meant the non-negative real value I/a; 9 + y 2 '* by * ts amplitude (am z, z=^tf), we mean the angle <p for which both cos(p = ~. p and sin cp = -py . When we calculate with absolute values, the rules 3, II, 1 4 hold unchanged, while 5. loses all meaning. Since we may accordingly operate, broadly speaking, in precisely the same ways with complex as with real numbers, by far the greater part of our previous investigations may be carried out in an entirely analogous manner in the realm of complex numbers, or transferred to the latter, as the case may be. The only considerations which will have to be omitted or suitably modified are those in which the numbers themselves (not merely their absolute values) are connected by the symbol < or >. In order to avoid repetitions, which this parallel course would otherwise involve, we have prefixed the sign to all definitions and theorems, from Chapter II onwards, which remain valid word for word when arbitrary real numbers are replaced by complex numbers, (this validity extending equally to the proofs, with a few small alterations which will be explained immediately). We need only glance rapidly over the whole of our preceding developments and indicate at each place what modification is required when we transfer them to the realm of complex numbers. A few words will also be said on the subject of the somewhat different geometrical representation. Definition 23 remains unaltered A sequence of numbers will now be represented by a sequence of points (each counted once or more than once) in the plane. If it is bounded (24, 1), none of its points lie outside a ciicle of (suitably chosen) radius K with origin at 0. Definition 25, that of a null sequence, and the theorems 26, 27, and 28 relating to such null sequences remain entirely unaltered. The sequences (z n ) with i (_;)n - are examples of null sequences whose terms are not all real. The student should form an exact idea of the position of the corresponding sets of points and prove that the sequences are actually null sequences. The definitions in 7 of roots, of powers in the general sense, and of logarithms were essentially based on the laws of order for real numbers. They cannot, therefore, be transferred to the realm of complex numbers in that form (cf. 55 below). The fundamental notions of the convergence and divergence of a sequence of numbers (39 and 40, l) still remain unaltered, 52. Complex numbers and sequences. 391 although the representation of z n -> now becomes the following l : If a circle of arbitrary (positive) radius e is described about the point as centre, we can always assign a (positive) number n Q such that all terms of the sequence (z n ) with index n > n Q lie within the given circle. The remark 39, 6 (1 st half) therefore holds word for word, provided we in- terpret the ^-neighbourhood of a complex number as being the circle mentioned above. In setting up the definitions 40, 2, 3, the symbols < and > played an essential part; they cannot, therefore, be retained unaltered. And although it would not be difficult to transfer their main content to the complex realm, we will drop them entirely, and accordingly in the complex realm we shall call every non-convergent sequence divergent ' 3 . Theorems 41, 1 to 12, and the important group of theorems 43, with the exception of theorem 3, remain word for word the same, together with all the proofs. The most important of these theorems were the Cauchy-Toeplitz limit-theorems 43, 4 and 5, and since we have in the meantime gained complete familiarity with infinite series, we shall formulate them once more in this place, with the extension indicated in 44, 10, and for complex numbers. Theorem 1. The coefficients of the matrix 221. (A) 20* are assumed to satisfy the two conditions: (a) the terms in each column form a null sequence, i.e. for every fixed n 0, as <x>. 1 For complex numbers and sequences, we preferably use in the sequel the letters *,(,,.... 2 We might say, in the case | * | -> + OO , that (z n ) is definitely divergent with the limit CO, or tends or diverges (or even converges) to oo. That would be quite a consistent definition, such as is indeed constantly made in the theory of functions. However, it evidently involves a small inconsist- ency relative to the use of the terms in the real domain, that e. g. the sequence of numbers ( l) n n should be called definitely or indefinitely convergent, according as it is considered in the complex or in the real domain. And even though, with a little attention, this may not give us any trouble, we prefer to avoid the definition here. 392 Chapter XII. Series of complex terms. (b) there exists a constant K such that the sum of the absolute values of any number of terms in any one row remains less than K, i. e. 9 for every fixed k ^> 0, and any n: Kol + KlH \~\ a kn\< K - Under these conditions, when (*, *i> ) zs an y nu ^ sequence, the numbers 00 *' = a kO *0 + a kl Z \ H = 2 a kn Z n=o also form a null sequence 9 . Theorem 2. The coefficients a kn of the matrix (A), besides satis- fying the two conditions (a) and (b), are assumed to satisfy the further condition 3 or> (c) 27 a kn --= A k -> 1 as k -> oo. n-O In this case, if z n +, we have also V = **0*0 + fl fcl *1+''' = iXn*,i -*f n=o (For applications of this theorem, see more especially 233, as well as 60, 62 and 63.) Unfortunately, we lose the first of the two main criteria of 9, which was the more useful of the two. Moreover, the proof of the second main criterion cannot be transferred to the case of complex numbers, as it makes use of theorems of order throughout. In spite of this, we shall at once see that the second main criterion itself in all its forms remains valid for complex numbers. The proof may be conducted in two different ways: either we reduce the new (complex) theorem to the old (real) one, or we construct fresh founda- tions for the proof of the new theorem, by extending the develop- ments of 10 to complex numbers. Both ways are equally simple and may be indicated briefly: 1. The reduction of complex sequences to real sequences is most easily accomplished by splitting up the terms into their real and imaginary parts. If we write z n = x n -\- iy n and f = f -(- irj, we have the following theorem, which completely reduces the question of the convergence or divergence of complex sequences to the corresponding real problem: 322. Theorem 1. The sequence (z n ) s= (x n + iy n ) converges to = -f- i v\ if, and only if, the real parts x n converge to and the imaginary parts y n converge to rj. 3 In consequence of (b), A k = Za kn is absolutely convergent and there- n fore, as the z n 's are bounded, by 41, Theorem 2, the series ^ai c n z n =i!s k 1S a l 80 absolutely convergent. n 52. Complex numbers and sequences. 393 Proof, a) If # n -*f and y n +rj, (x n f) and (y n - 77) are null sequences. By 26,1, the same is true of i(y n rj) and, by 28,1, of ( x n -") + i(y n ~~ y}> l - e - f ( z n ) b) If z n **, |2 OT I is a null sequence; since 4 (x n f) and (y n rf) are also null sequences, by 26, 2, i. e. we have both The theorem is established. The theorem at which we are aiming follows immediately: Theorem 2. For the convergence of a complex sequence (z w ), the conditions of the second main criterion 47 are again necessary and sufficient, namely, that, for every choice of e > 0, we should be able to assign n so that for every n >> n Q and every n' > n Q . Proof, a) If (2 n ) converges, so do (x n ) and (y w ) by the preceding theorem As these are real sequences, we may apply 47, and, given e > 0, we may choose n l and n 3 so that I x n' x n I < o f r ever y n > n i an d every n' > n t , and I ^ y n | < TT for every n > w 9 and every n' > n^ Taking n Q greater than n x and n 2 , we have accordingly, for every n > n Q and every w' > n , The conditions of our theorem are therefore necessary. b) If, conversely, (z n ) fulfils the conditions of the theorem, i. e. given e > 0, we can determine n so that | z n ' z n \< e, provided only that n and n' are both > n , we have also, for the same n and n' (by our last footnote) *n'-*< e and - <' * We have in general since 394 Chapter XII. Series of complex terms. By 47, this implies that (x n ] and (yj are convergent, so that (z n ) must also converge, by the preceding theorem; the conditions of our theorem are therefore also sufficient. 2. Direct treatment of complex sequences. In the treatment of real sequences, nests of intervals constituted our most frequent resource. In the complex domain, nests of squares will render us the same services: 223. Definition. Let Q , Q lf @ 3 , ... denote squares, whose sides will for simplicity be assumed parallel to the coordinate-axes. If each square is entirely contained in the preceding and if the lengths I 0> l lt ... of the sides form a null sequence, we shall say that the squares form a nest. For nests of squares, we have the Theorem. There exists one and only one point belonging to all the squares of a given nest of squares. (Principle of the innermost Point.) Proof. Let the left hand bottom corner of Q n be denoted by n + ta anc * me right hand upper corner by b n -}-ib*. A point z = x -f- i y belongs to the square Q n if, and only if 5 , Now, in consequence of our hypotheses, the intervals J n = a n . . . b n on the a;- axis, and similarly the intervals /n = n*...&n* on the y-axis, form a nest of intervals. There is therefore exactly one point f on the #-axis and exactly one point iv\ on the y-axis belonging to all the intervals of the corresponding nest. But this means that there is also exactly one point f = f + 117, belonging to all the squares Q n . We are now in a position to transfer definition 52 and theorem 54 to the complex domain: 224. Definition. // (z n ) is an arbitrary sequence, is said to be a limiting point or point of accumulation of the sequence if, given an arbitrary 6 > 0, the relation k-l< is satisfied for an infinity of values of n (in particular, for at least one n > any given n Q ). 225. Theorem. Every bounded sequence possesses at least one limiting point. (Bolzano- Weierstrass Theorem.) Proof. Suppose \z n \ < K and draw the square Q Q whose sides lie on the parallels to the axes through & an d *% AH them's 5 This statement at the same time expresses, in pure arithmetical lang- uage, the relations of magnitude framed in geometrical form in the theorem and definition 228. 52. Complex numbers and sequences. 395 are contained in it, i. e. certainly an infinity of z n 's. (> is divided by the cooi dinate axes into four equal squares One at least of the four must contain an infinity of z n 's. (In fcict, if there were only a finite number in each, there would also be only a finite number in @ , which is not the case) Let Q denote the first quarter' 1 which has this property. This we again proceed to divide into four equal squares, denoting by @ 3 the first quarter which contains an infinity of points z n , and so on. The sequence Q , Q , Q 2 , ... forms a nest of squares, since each Q n lies within the preceding and the lengths of the sides form a null sequence, namely (2K-^-J. Let f denote the innermost point of this nest 7 ; f is a point of accumulation of (z n ). For if e is given > and m is chosen so that the side of Q m is less than -|-, the whole of the square Q m lies within the 6- neighbourhood of , and, with it, an infinite number of points z n also lie in this neighbourhood. Therefore f is a point of accumulation of (z n ) 9 and the existence of such a point is established. The validity of the second main criterion for the complex domain, i. e. of the theorem 222, 2, formulated above may now be established once more, but without any appeal to the "real" theorems, on the same lines as in 47. Proof, a) If z n >, i.e. (z n f ) is a null sequence, we can -C< and *'-f<- determine n Q so that provided only that n and n' are simultaneously > n [see part a) of the proof of 47]. For these ns and w"s, we therefore also have The condition is accordingly necessary. b) If, conversely, the e- condition is fulfilled, (z n )is certainly bounded. In fact, if m > n Q and n > m, \. e. every Z M with n "> m lies in the circle of radius e round z " n Taking K to be larger than all the m numbers \z^\ 9 |* 3 |, . . *-* * WC 6 We regard the four quarters as numbered in the order in which the four quadrants of the xy -plane are habitually taken. 7 The process of obtaining this point corresponds exactly to the method of successive bisection so often applied in the real domain. 396 Chapter XII. Series of complex terms! By our preceding theorem, it follows that (zj tas at least one limiting point . Supposing there exists a second limiting poi&J ' + f, choose / 6 _JL|r_ t | < e 3 K ; l which is positive. By 224, the definition of limiting point, we can choose n Q as large as we please and yet have an n > n for which | z n | < e and also an ri > n for which | z n > f ' | < Thus above any number w , however large, there exist a pair of indices n and w' for which 8 Iv -*! > This contradicts our hypothesis. Accordingly f must be the unique limiting point, and outside the circle of radius e round f there is only a finite number of points z n . If n is suitably chosen, we there- fore have \z n |<e for every n > n , and consequently z n *. The condition of the theorem is therefore sufficient also 9 . 53. Series of complex terms. As a series 2 a n of complex terms must obviously be interpreted as the sequence of its partial sums, the basis for the extension of our theory of infinite series has already been provided by the above. Corresponding to 222, 1, we have first the 226. Theorem. A series 2 a n of complex terms is convergent if, and only if, the series 2 9t (a n ) of the real parts of its terms and the series 2 $ (flj of their imaginary parts converge separately. Further > if these two series have the sums s' and s" respectively, the sum of 2 a n is s = s' + ; 5 ". In accordance with 222, 2 the second principal criterion (81) for the convergence of infinite series remains unaltered in all its forms, and, at the same time, the theorems 83 deduced from it, on the algebra of convergent series, also retain their full validity. Since, in the same way, theorem 85 also remains unchanged, we shall, as before, distinguish between absolute and non-absolute con vergence of series of complex terms (Def. 86). hence 9 Hence we may also say: (z n ) converges if, and only if, it is bounded and possesses only one point of accumulation. This is then at the same time the limit of the sequence. 53. Series of complex terms. 397 Here again we have the Theorem. The series 2 a n of complex terms is absolutely con- 227, vergent if, and only if, both the series 2^i(a n ) and 2%(a n ) are ab- solutely convergent. The proof results simply from the fact that every complex number z = x-\-iy satisfies the inequalities (cf. p. 393, footnote 4) In consequence of this simple theorem, it is at once clear that, with series of complex terms as with real series, the order of the terms is immaterial if the series converges absolutely (Theorem 88,1). If, however, 2a n is not absolutely convergent, either 2$l(a n ) or ^3(0 must be conditionally convergent. By a suitable rearrangement of the terms, the convergence of the series 2a n may therefore be des- troyed in any case, as in the proof of theorem 89, 2, that is: In the case of series of complex terms also, the convergence, when it is not absolute, depends essentially on the order of succession of the terms. (Regarding the extension to series of complex terms of Riemanns rearrangement theorem 44, cf. the remarks on the following page.) The next theorems, 89, 3 and 4, as also the main rearrangement theorem 90, which relate to absolutely convergent series, still remain valid, without modification or addition, for series of complex terms. Since the determination of the absolute convergence of a series is a question relating to series of positive terms, the whole theory of series of positive terms is again enlisted for the study of series of complex terms: Everything that was proved for absolutely convergent series of real terms may be utilized for absolutely convergent series of complex terms If we omit power series from consideration for the present, we observe, on looking over the later sections of Part II ( 18 27), that the developments of Chapter X are the first for which there is any question of transference to series of complex terms. Abel's partial summation 182, being of a purely formal nature, and its corollary 183, of course hold also for complex numbers, and so does the convergence- test 184 which was based directly on them. The special forms of this test may also all be retained, provided we keep to the convention agreed on in 220, 5, in accordance with which all sequences assumed to be monotone are real. In the case of du Bois- Reymond's and Dedekind's tests, even this precaution becomes unnecessary: they hold word for word and without any restriction for arbitrary series of the form 2 a n b n , with complex a n and b n . Riemanris rearrangement theorem ( 44) is, on the contrary, essen- 398 Chapter XII. Series of complex terms. tially a "real" theorem. In fact, if a series 2a n of complex terms is not absolutely convergent, so is one at least of the two series JJjR^iJ and 2%(a n ), by 227. By a suitable rearrangement, we can therefore, in accordance with Riemanns theorem, produce in one of these two series a prescribed type of convergence or divergence. But the other one of the two series will be rearranged in precisely the same manner, and there is no immediate means of foreseeing what the effect of the rearrangement on this series or on 2a n itself will be. It has recently been shown, however, that if 2a n is not absolutely convergent, it may be transformed by a suitable rearrangement into a series, again con- vergent, whose sum may be prescribed to have either any value in the whole complex plane or any value on a particular straight line in this plane, according to the circumstances of the case 10 . The theorems 188 and 189 of Mertens and Abel on multi- plication of series ( 45) again remain valid word for word, together with the proofs. For the second of these theorems we must, it is true, rely on the second proof (Cesaro's) alone, as we have provisionally skipped the consideration of power series (cf. later 232). At this point we are in possession of the whole machinery required for the mastery of series of complex terms and we can at once proceed to the most important of its applications. Before doing so, however, we shall first deduce the following extremely far-reaching criterion. 00 228. Weierstrass* criterion u . A series 2 a n of complex terms, for which with A n bounded, where <x is complex and arbitrary, and 12 A>1, 10 We thus have the following very elegant theorem, which in a certain sense completes the solution of the rearrangement problem: The "range of summation" of a series Z a n of complex terms i. e. the set of values which may be obtained as sums of convergent rearrangements of Ea n is either a definite point, or a definite straight line, or the entire plane. Other cases cannot occur. A proof is given by P. L&vy (Nouv. Annales (4), Vol. 5, p. 506, 1905), but an unexception- able statement of the proof is not found earlier than in E. Steinits (Bedmgt kon- vergente Reihen und konvexe Systeme, J. f. d. reine u. angew. Math., Vol. 143, 1913; Vol. 144, 1914; Vol. 146, 1915). For the (more restricted) result that every conditionally convergent series Sa n ** s can be rearranged to give another convergent series Sa n f =* s' with s' =t= s 9 W. Threlfall has given a fairly short proof (Bedingt konvergente Reihen, Math. Zschr., Vol. 24, p. 212, 1926). 11 J. f. d. reine u. angew. Math., Vol. 51, p. 29, 1866; Werke I, p. 185. 12 An equality of this kind may of course always be assumed ; we need only write A n = H* ( 1 " ' ) as a definition. What is essential in the condition \ n a n / is here, as previously (cf. footnote to 166), that when a and A are suitably chosen the An* should be bounded. It is substantially the same thing to assume that ! + //! + B n /n* wjth A > 1 and B n bounded. 53. Series of complex terms. 399 is absolutely convergent if, and only if, SR (a) > 1 . For JR (a) < the series is invariably divergent. If 0<8fl(a)<^l, both the series n=o are convergent. Proof. 1. Let cc = (t-\-iy and let us first assume /?== $ In that case, if | A n \ < K, say, we write, as is permissible, > 1. and it follows at once that, if /?' is any number such that 1 < ^^. JL a n \ n for every sufficiendy large n. By Raabes test, the series -^|a n | is therefore convergent. 2. Now suppose SH() = /?^1- In that case, since for sufficiently large values of n, it follows from Gauss's test 172 that 2\a n \ is divergent. 3 a. If, on the other hand, 9ft (a) = ft < 0, our last inequality shows that then Therefore 2 a n must now diverge. 3b. If $K(a) = = 0, i. e . f > it is easy to verify that we then have where H > 1 and is the smaller of the two numbers 2 and Jl, and ^'s are again bounded. Accordingly, if c denotes a suitable constant, 13 As regards the series 2 a n itself, it was shown by Weierstrass, I.e., that this is also divergent whenever Ot (a) fS 1. The proof is somewhat troublesome. A further more exact investigation of the series 27 a n itself in the case 5^ 0? (a) fg 1 is given by A. Pringsheim (Archiv d. Math, und Phys. (3), Vol. 4, pp. 1 19, in particular pp. 13 17. 1902), J. A. Gmeiner, Monatshefte f. Math. u. Phys., Vol. 19, pp. 149103. 1908. 400 Chapter XII. Series of complex terms. for every n ^> w, say. It follows by multiplication that "n n-1 >n i--- Hence | a n \ > C m -\ a m \ , for every n>m, and w cannot tend to 0, so that 2a n again diverges (cf. 17O, 1). 4. If, finally, , we have to show that both the series are convergent. Now as in 1. we have, for every sufficiently large n, I a . , 8' * + 1 < 1 , with </?'</?, | a n n r f so that | a n | diminishes monotonely from some stage on, and there fore tends to a definite limit ^> 0. Accordingly, a) the series J"(|a n | |# n + i|) is convergent, by 131, and has, moreover, all its terms positive for sufficiently large n's. Now 1- = V since the fraction on the right hand side tends to the positive limit ~~ when n *-f-c that on the left is, for every sufficiently large n, less than a suitable constant A. By 70, 2, this means that 2\a n w . hl | converges with2(| n | |0 w +i|)- We can show more precisely, how- ever, that b) a n + 0. For it again follows, by multiplication, from that (n ^ The right hand side (by 126, 2) tends to as n (cf. 17O, 1) we must have fl n >0. Now the series hence *=o is a sub series of 2(a n n+1 ) and therefore converges absolutely, by a); also, since | a n \ + | n + 1 1 *0 with a n , we may omit the brackets, by 83, supplement to theorem 2. This proves the con- vergence of 2 ( l) n a n . This theorem enables us to deduce easily the following further theorem, which will be of use to us shortly: 54. Power series. Analytic functions. 401 Theorem. //, as in the preceding theorem, 229. _?i. i _ _!i _ *. I a arbitrarv > ^ ~~a"7 ~n n*> I (4) bounded, the series 2a n z n is absolutely convergent for \z\ < 1, divergent for every | z | > 1 , and for the points of the circumference \ z \ = 1 , the series will a) converge absolutely, if *3i () > 1 , b) converge conditionally, if 0< 3t(a)<^l, except possibly for the single point z = + 1 c) diverge, if 9R (a) ^ . Proof. Since the statements relative to \z\^\ are immediately verified. For | z | = 1 , the statement a) is an immediate consequence of the con- vergence of -2|0 n | ensured by the preceding theorem. Similarly c) is an immediate consequence ,of the fact established above, that in this case | a n \ remains greater than a certain positive number for every sufficiently large n. Finally, if < 5R(a) ^ 1 and z 4* -f- 1, the convergence of 2a n z n follows from Dedekind's test 184, 3. For we proved in the preceding theorem that 2\a n a n ^. 1 \ converges and a n >0; that the partial sums of 2z n are bounded, for every (fixed) 24>-|-l on the circum- ference | z | = 1 , follows simply from the fact that for every n II-*! 54. Power series. Analytic functions. The term "power series" is again used here to denote a series of the form 2a n z n , or, more generally, of the form 2a n (z z^ n 9 where now both the coefficients a n and the quantities z and Z Q may be complex. The theory of these series developed in 18 to 21 remains vab'd without any essential modification. In transferring the considerations of those sections, we may therefore be quite brief. Since the theorems 98, 1 and 2 remain entirely unaltered in the new domain, the same is true of the fundamental theorem 93 itself, on the behaviour of power series in the real domain. Only the geo- metrical interpretation is somewhat different: The power scries 2a n z* 14 If we take into account Pnngsheim's result mentioned in the preceding* footnote, we may state here, more definitely except for * = -fl. 40 2 Chapter XII. Series of complex terms. converges indeed absolutely for every z interior to the circle of radius r round the origin 0, while it diverges for all points outside that circle. This circle is called the circle of convergence of the power series and the name radius applied to the number r thus becomes, for the first time, completely intelligible. Its magnitude is given as before by the Cauchy-Hadamard theorem 94. Regarding convergence on the circumference of the circle of con- vergence, we can no more give a general verdict than we could re- garding the behaviour at the endpoints of the interval of convergence in the case of real power series. (The examples which follow immedia- tely will show that this behaviour may be of the most diverse nature.) The remaining theorems of 18 also retain their validity unaltered. 230. Examples. 1. 2z n ] r=l. In the interior of the unit circle, the series is convergent, with the sum . On the boundary, i. e. for | z \ = 1 , it is everywhere di- L Z vergent, as z n does not -* there. z n 2. Jf? 5- ; f = 1. This series l5 remains (absolutely) convergent at all the boundary points | z \ = 1 . z n 8. 5] ; r=l. The series is certainly not convergent for all the ft boundary points, for* = l gives the divergent series . However, it is also not divergent for all these points, since z = 1 gives a convergent series. In fact, theorem 229 of the preceding section shows, more precisely, that the series must converge conditionally at all points of the circumference |jr| = l different from + 1 ; for we have here The same result may also be deduced directly from Dinchlefs test 184, 2, since Z z n has bounded partial sums for z =f= -f 1 and | s | = 1 (cf. the last formula of the preceding section) and tends monotonely to 0. As n z n the convergence can, however, only be conditional 16 . 4. 5? -. ; r = 1 . This series diverges at the four boundary points 4 n and t, and converges conditionally at every other point of the boundary. 16 If 2a n z n has real coefficients (as in most of the subsequent examples) this power series of course has the same radius as the real power series 2a n x n . 16 These facts regarding convergence may also be deduced from 185, 5, by splitting up the series into its real and imaginary parts. Conversely, how- ever, the above mode of reasoning provides a new proof of the convergence of these two real series. 54. Power series. Analytic functions. 403 z n 5. For , r = -f-OO. For 2'n!* n , r=0; thus this series converges nowhere but at z=0. 6. The series J (-!)-. and ^(_l)*-^__ are everywhere convergent. 7. A power series of the general form 2a n (z z ) n converges absolutely at all interior points of the circle of radius r round z ot and diverges outside this circle, where r denotes the radius of 2a n z n . Before proceeding to examine the properties of power series in more detail, we may insert one or two remarks on Functions of a complex variable. If to every point z within a circle $ (or more generally, a domain 17 ) a value w is made to correspond in any particular manner, we say that a junction w = f(z) of the complex variable z is given in this circle (or domain). The correspondence may be brought about in a great number of ways (cf. the corresponding remark on the concept of a real function, 19, Def. l) ; in all that follows, however, the func- tional value will almost always be capable of expression by an explicit formula in terms of z, or else will be the sum of a convergent series whose terms are explicitly given. Numerous examples will occur very shortly; for the moment we may think of the value w, for instance, which at each point z within the circle of convergence of a given power series represents the sum of the series at that point. The concepts of the limit, the continuity, and the differentiability of a function are those which chiefly interest us in this connection, and their definitions, in substance, follow precisely the same lines as in the real domain : 1. Definition of limit. If the function w=f(z) is defined 18 for 231. every z in a neighbourhood of the fixed point , we say that iim f(z] = co or _ f(z)-+a> for *-*, 17 A strict definition of the word "domain" is not needed here. In the sequel, we shall always be concerned with the interior of plane areas bounded by a finite number of straight lines or arcs of circles, in particular with circles and half-planes 18 f(z) need not be defined at the point itself, but only for all z's which satisfy the condition 0< |* | <Q. The d of the above definition must then of course be assumed 404 Chapter XII. Series of complex terms. if, given an arbitrary e > , we can assign d d (e) > so that \f(z)-a>\<e for every 2 satisfying the condition < | z \ < 6; or which comes to exactly the same thing 19 if for every sequence (z n ) converging to , whose terms lie in the given neighbourhood of and do not coincide with , the corresponding functional values w n = f(z n ) con- verge to co. If we consider the values of f (z) , not at all the points of a neigh- bourhood of , but only at those which lie, for instance, on a parti- cular arc of a curve ending at , or in an angle with its vertex at , or, more generally, which belong to a set of points M, for which is a point of accumulation, we say that limf(z) = co or f(z)*-co as z+> along that arc, or within that angle, or in that set M, if the above conditions are fulfilled, at least for all points z of the set M which come into consideration in the process. 2. Definition of continuity. If the function w = f(z) is defined in a neighbourhood of and at itself, we say that f(z) is continuous at the point , if lim f(z) exists and is equal to the value of the function at , i. e. if f(z) */"() We may also define the continuity of f(z) at when z is restricted to an arc of a curve containing the point , or an angle with its vertex at , or any other set of points M that contains and of which is a limiting point; the definitions are obvious from 1. 3. Definition of differentiability. If the function w = f(z) is de- fined in a neighbourhood of and at itself, f(z) is said to be differ- entiate at , if the limit lim ' exists in accordance with 1. Its value is called the differential coeffi- cient of f(z) at and is denoted by /"(). (Here again the mode of variation of z may be subjected to restrictions.) We must be content with these few definitions concerning the general functions of a complex variable. The study of these functions in detail constitutes the object of the so-called theory of functions, one of the most extensive domains of modern mathematics, into which we of course cannot enter further in this place 30 . 19 Same proof as in the real domain. 80 A rapid view of the most important fundamental facts of the theory of functions may be obtained from two short tracts by the author : Funktionen- 54. Power series. Analytic functions. 405 The above explanations are abundantly sufficient to enable us to transfer the most important of the developments of 20 and 21 to power series with complex terms. In fact, those developments remain valid without exception for our present case, if we suitably change the words "interval of conver- gence' to "circle of convergence" throughout Theorem 5 (99) is the only one to which we can form no analogue, since the concept of integral has not been introduced for functions of a complex argument. All this is so simple that the reader will have no trouble, on looking through these two sections again, to interpret them as if they had been intended from the first to relate to power series with complex terms. At the most, a few remarks may be necessary in connection with Abel's limit theorem 100 and theorem 107 on the reversion of a power series. In the case of the latter, the convergence of the series y + As y' 2 H ---- > an( ^ hence of the series y -f- b^ y 1 -\ ---- , which satis- fied the conditions of the theorem, were only proved for real values of y. This is clearly sufficient, however, as we have thereby proved that this power series has a positive radius of convergence, which is all that is required. As regards Abel's limit theorem, we may even corresponding to the greater degree of freedom of the variable point z prove more than before, and for this reason we will go into the matter once more: Let us suppose 2 a n z n to be a given power series, not everywhere convergent, but with a positive radius of convergence. We first observe that, exactly as before, we may assume this radius = 1 without intro ducing any substantial restriction On the circumference of the circle of convergence, | z \ = 1, we assume that at least one point z exists at which the series continues to converge. Here again we may assume that Z Q is the special point + 1. In fact, if z 4 s + 1> we need only put "n*0* =*,!> the series 2a n ' z n also has the radius 1 and converges at the point -j- 1. The proof on^mally given, where everything may now be interpreted as "complex", then establishes the Theorem. // the power series 2' a n z n has the radius 1 and remains convergent at the point -f- 1 of the unit circle, and if 2 a n = s t then we also have if z approaches the point + 1 along the positive real axis from the origin 21 0. theorie, I. Teil, Grundlagen der allgemeinen Theorie, 4 th ed., Leipzig 1930; II. Teil, Anwendungen und Weiterfiihrung der allgemeinen Theone, 4 th ed., Leipzig 1931 (Sammlung Goschen, Nos. 668 and 703). 21 We are therefore dealing with a limit of the kind mentioned above in 231, 1. 14 (051) 406 Chapter XII. Series of complex terms. 233. We can now easily prove more than this: Extension of AbeVs theorem. With the conditions of the preceding theorem, the relation remains true if the mode of approach of z to +1 is restricted only by the condition that z should remain within the unit circle and in the angle between two arbi- trary (fixed) rays which pene- trate into the interior of the unit circle , starting from the point + 1 (see Fig. 10). The proof will be con- ducted quite independently of previous considerations, so that we shall thus obtain a third proof of Abel's theorem. Let Z Q , k , . be Fig. 10. We have to show that any sequence of points of limit -\- 1 in the described portion of the unit circle fto- if, as before, we write 2a n z n = f(z). In Toeplitz theorem choose for a ftn the value and apply the theorem to the sequence of partial sums s n = "o + i H !-> which, by hypothesis, converges to s. It follows immediately that - **)<* = ( ] n=0 also tends to s as k increases. This proves the statement, provided we can show that the chosen numbers a kn satisfy the conditions (a), (b) and (c) of 221. Now (a) is clearly fulfilled, as * fc 1, and the CO sum of the & th row is now A k = (l z k ] ^z k = 1, so that (c) is n=o fulfilled. Finally (b) requires the existence of a constant K such that for all points z = z k 1 in the angle (or any sector-shaped portion of it with its vertex at +1). It only remains, therefore, to establish 54. Power series. Analytic functions. 407 the existence of such a constant. This reduces (v. Fig. 10) to proving the following statement: If z = l Q (cos <p + * s i n 9?) w#A | <P | ^ <P < ^ and < Q <I Q Q < 2 cos <p , a constant A = A (<^ , ^ ) exists, depending only on <p and Q O , such that ^^ A for every z of the type described. In the proof of this statement, it is sufficient to assume Q Q = COS<PQ> an d in that case we may at once o show that A = is a constant of the desired kind. In fact, the cos<p statement then runs: 1 - yi - 2 Q cos <p + e a cos <Po or 2 Q cos 9? + 2 <I cos 9? + Q* cos 2 <p , for < Q<L cos9? and \<p\ <; go Q . By replacing 99 by <p and 2 by cos<p on the left hand side, the latter is increased; therefore it cer- tainly suffices to show that q cos 9 ^ q cos <p + ^ <P cos 2 9 , - which is obviously true. This extension of Abel's theorem to "com- plex modes of approach" or "approach within an angle" is due to O. Stolz 22 . This completes the extension to the case of complex numbers of all the theorems of 20 and 21 with the single exception of the theorem on inte- gration, which we have not defined in the present connection. In particular, it is thereby established that a power series in the interior of its circle of convergence defines a function of a complex variable, which is continuous and different iable the latter "term by term" and as often as we please in that domain, and accordingly possesses the two properties which above all others are required, in the case of a function, for all purposes of practical application. For this reason, and on account of their great importance in further developments of the theory, a special name has been reserved for functions representable in the neighbourhood of a point 28 Zeitschrift f. Math. u. Phys., Vol. 20, p. 369, 1875. In recent years the question of the converse of Abel's theorem has been the object of numerous investi- gations, i. e. the question, under what (minimum of) assumptions relating to the coefficients a n , the existence of the limit of J (z) as z > 1 (within the angle) entails the convergence of a n . An exhaustive survey of the present state of research in this respect is given in papers by G. H . Hardy and J. E. Ltttlewood, Abel's theorem and its converse, Proc. Lond. Math. Soc. (2), I. Vol. 18, pp. 205 235, 1920; II. Vol. 22, pp. 254269, 1923; III. Vol. 26, pp. 219 23G, 1926. Cf. also theorems 278 and 287. 408 Chapter XII. Series of complex terms. z by a power series E a n (z z Q ) n . They are said to be analytic or regular at Z Q . By 99, such a function is then analytic at every other interior point of the circle of convergence; it is therefore said simply to be analytic or regular in this circle 23 . In particular, a series everywhere convergent re- presents a function regular in the whole plane, which is therefore shortly called an integral function. All the theorems which we have proved about functions expressed by power series are theorems about analytic functions. Only the two fol- lowing, which are of special importance in the sequel, need be expressly formulated again. 234. 1- If two functions are analytic in one and the same circle , then so are (by 21) their sum, their difference, and their product. For the quotient the corresponding statement is primarily true (by 105, 4) only if the function in the denominator is not zero at the centre of the circle, and provided, if necessary, that this circle is replaced by a smaller one. 2. If two functions, analytic in one and the same circle, coincide in a neighbourhood, however small, of its centre (or indeed at all points of a set having this centre as point of accumulation), the two functions are completely identical in the circle (Identity theorem for power series 97). Besides stating these two theorems, which are new only in form, we shall prove the following important theorem, which gives us some information on the connection between the moduli of the coefficients of a power series and the modulus of the function it represents : 00 235. Theorem. If f(z) = Z a n (z z Q ) n converges for \ z Z Q \ < r, then = o \^\^ M P (P = 0, 1, 2, . . .), G if < < r and M ~ M (Q) is a number which \ f(z) \ never exceeds along the circumference \ z Z Q \ = Q. (Cauchy's inequality.) Proof 24 . We first choose a complex number 77, of modulus = 1, for which however -rf 1 3= 1 for any integral 25 exponent q ^ 0. Now we consider the function *(*) = (*- *o) k 28 A function is accordingly said to be "analytic" or "regular" in a circle ,ft when it can be represented by a power series which converges in this circle. 24 The following very elegant proof is due to Weierstrass (Werke II, p. 224) and dates as far back as 1841. Cauchy (Me"moire lithogr., Turin 1831) proved the formula indirectly by means of his expression for / (z) in the form of an integral. The existence of a constant M that | f(z) \ never exceeds on | z z \ 2 is practically obvious, of course, since M = 2 \ a n \ Q n clearly has this property. This Mis obviously also such that | a n \ Q n ^ M. But the above theorem states that every M that | / (z) \ never exceeds has the property that | a n \ Q n is always 5^ M. 25 Such numbers 17 of course exist, for if t\ cos (a TT) + i sin (a TT), then if* cos (q a TT) + i sin (q a TT) ; this is never 1 if a is chosen irrational. 54. Power series. Analytic functions. 409 for a specific integral value of the exponent k ^ and an arbitrary constant coefficient a. If we denote by g Q , g l9 g >2 , ... the values of this function for z = Z Q + Q- rj v , v = 0, 1 , 2, . . . , we have for n ^ 1 fc 1 *i kn hence I The expression on the right hand side contains only constants, besides the denominator n; it therefore follows that the arithmetic mean \-gn-l ~ as n increases. In the case k 0, we should be concerned with the identically constant function g(z) s= a, for which since the ratio is equal to a for every n, in this case. If we consider the rather more general function ^)-t? + ^^^ where I and m are fixed integers ^> 0, and now form the arith- metic mean ----- hgn-l (where, as before, g v = g (z -f- Q rj v ), v = 0, 1, . . .), this clearly + b , by the two cases just treated. If, further, it is known that the function g(z)> for every z of the circumference | z Z Q \ = Q, is never greater than a certain constant K, we have also gp-f ffi + " + gn-l ^ n A" ^ W ~~ 7J and therefore also With these preliminary remarks, the proof of the theorem is now quite simple: Let p be a specific integer ^0. As 2j\ a n \Q* con verges, given e > 0, we can determine q > p so that 410 Chapter XII. Series of complex terms. A fortiori, we then have for all values of z such that | z Z Q \ = Q , I !(*-*o) B l< nq + l and therefore, for the same values of z 9 n=0 if M has the meaning given in the text. Accordingly, on the circum- ference | Z ZQ | = Q , The function between the modulus signs is of the kind just considered. The inequality | b \ ^ K there obtained now becomes , K^tf I **p 1 === p * and, as e was arbitrary and > 0, we have, in fact, (cf. footnote to 41, l) I I < q. e. d. 55. The elementary analytic functions. I. Rational functions. 1. The rational function w = - is expressible as a power series for every centre Z Q + + 1 : 1 1 11 S 1 / ..xi. l-z I-Z Q -(Z-Z O ) I~ 1~ and this series converges for [ z z | < 1 1 z \ i. e. for every z nearer to z than -f- 1; in other words, the circle of convergence of the series is the circle with centre Z Q passing through the point +1. The function 1 - is thus analytic at every point different from +1. With reference to this example, we may briefly draw attention to the following phenomenon, which becomes of fundamental importance in the theory of functions: If the geometric series 2z n , whose circle of convergence is the unit circle, is expanded by Taylor's theorem about a new centre z within the unit circle, we could assert with certainty, by that theorem, that the new series converges at least in the circle of centre z l which touches the unit circle on the inside. We now see that the circle of convergence of the new series may very possibly extend beyond the boundary of the old. This will always, be the case, in fact, when z t is not real and positive. If z t is real and negative, the new circle will indeed include the old one entirely. (Cf. footnote to 99, p 176.") 55. The elementary analytic functions. I. Kational functions. 411 2. Since a rational integral function <*o + a i z + a * z *-\ ----- 1" a m z m may be regarded as a power series, convergent everywhere, such functions are analytic in the whole plane. Hence the rational functions of general type are analytic at all points of the plane at which the denominator is not 0, i. e. everywhere, with the exception of a finite number of points. Their expansion in power series at a point z , at which the denominator is 4 s 0> * s obtained as follows: If z is replaced by z o + ( z *o) both in the numerator and denominator of such a function, these being then rearranged in powers of (z * ), the function takes the form where, on account ot our assumption, & ' 4. 0. We may now carry out the division in accordance with 105, 4 and expand the quotient in the required power series 26 of the form Sc n (s #o) n * II. The exponential function. The series is a power series converging everywhere, and therefore defines a func- tion regular in the whole plane, i. e. an integral function. To every point z of the complex plane there corresponds a definite number w, the sum of the above series. This function, which for real values of z has the value e z as de- fined in 33, may be used to define powers of the base e (and then further those of any positive base) for all complex exponents: 80 An alternative method consists in first splitting 1 up the function into partial fractions. Leaving out of account any part which represents a rational integral function, we are then concerned with the sum of a finite number of fractions of the form A A i 1 \tf each of which we may, by 1, expand separately in a power series of the form 2c n (g z )*, provided z 4= a. This method enables us to see, moreover, that the radius of the resulting* expansion will be equal to the distance of z from the nearest point at which the denominator of the given function vanishes. 412 Chapter XII. Series ol complex terms. 236. Definition. For all real or complex exponents, the meaning to be attributed to the power e* is defined, without ambiguity, by the relation And if p is any positive number, p z shall denote the value determined, without ambiguity, by the formula where log/> is the (real) natural logarithm of p as defined 21 in 36. (For a non-positive base b, the power b z can no longer be uniquely defined; cf., however, 244.) As there was no meaning attached per se to the idea of powers with complex exponents, we may interpret ihom in any manner we please. Reasons of suitability and convenience can alone determine the choice of a particular interpretation. That the definition just given is a th )r- oughly suitable one, results from formula 91, example 3 (leaving out of account the obvious requirement that the new definition must coincide with the old one for real values of the exponent 28 ); this formula was proved by means of a multiplication of series, the validity of which holds equally for real and complex variables and the formula must accordingly also hold for any complex exponent; it is 237. ei e z * = e i+* whence also This important fundamental law for the algebra of powers therefore certainly remains true. At the same time it provides us with the key to the further study of the function e z . 238. 1. Calculation of e z . For real y's, we have = cos y -f- i sin y . 27 It may be noted how far removed this definition is from the elementary definition lt x k is the product of k factors all equal to x". At first sight, there is no knowing" what value belongs e. g. to 2 l ; yet this value is in any case uni- quely determined by the above definition. 28 By 234, 2, there can exist no other function than the function a 2 just defined which is regular in the neighbourhood of the origin and coincides on the real axis z = x with the function e x defined by 33. For this reason we may indeed say that every definition of e z differing from the above would necessarily be unsuitable, 55. The elementary analytic functions. 11. The exponential function. 413 Hence it follows that, for z ~ x + i y> e* = e* + iv -== e x e iv e r (cos j/ By means of this lormula 20 the value of e z may easily be determined for all complex z's. This formula enables us, besides, to obtain in a convenient and complete manner an idea of the values which the function e z assumes at the various points of the complex plane (in short, of its stock of values}. We note the following facts. 2. We have \e z \ = **<*> = e*. In fact \e**\ = (cosy -f isiny | Vcos 2 y -j-sin s y = 1, hence | e z \ = \ e x \ - \ e iy \ = e x , because e x > and the second factor = 1. Similarly, am e z = 3 (z) = y, also from the formula 238, 1 just used. 3. e z has the periods 2kni, that is to say, for all values of z, e* = e *+**\ = e z *- 2 * w <, (k ^0, integral). For if we increase z by 2 71 i its imaginary part y increases by 2ji, \\ hile its real part remains unaltered, and by 1. and 24, 2, this leaves the value of the function unchanged. Every value which e z is able to assume accordingly occurs in the strip n < 3 (z) = y <i n, or in any strip which may be obtained from it by a parallel translation. Every such strip is called a period-strip; Pig. 11 represents the first- named of these strips. 4. e z has no other period, in- O deed, more precisely: if between two special numbers z and z^ we have the relation Fig. 11. this necessarily implies that For we first infer that e z *~~ z * = 1, then we note that if e z = e x + iv = ^(cos y -f- isiny) = 1, ae Euler: Intr. in Analysin inf. Vol. T, 138. 1748. 14* (G51) 414 Chapter XII. Series ot complex terms. we must by 2. have *=!, hence x = 0. Further, we also have cos y -f- i sin y = 1 i. e. cos y = 1 , sin y = 0, hence y = 2 k n . Thus, as asserted, 5. e* assumes every value w 4= owc0 awd only once in the period strip; or: the equation e z = w lt for given ^=^0, has one and only one solution in that strip. If w^ = R^ (cos ^ + * sin &J with # t > 0, the number is certainly a solution of e 3 = w t , as e zi = 6 \wRi e i'^ = R^ ( cos By 3., the numbers (ft = 0, 1, 2,...) are also solutions of the same equation, and by 4. no other solutions can exist. Now k may always be chosen, in one and only one way, so that n < 3 (^ -f 2 k n i} <^ + n, q. e. d. 6. The value is never assumed by e z \ for, by 237, e z.e- z = \, so that e z can never be 0. 7. The derivative (e z )' of e z is again e z y as follows at once by differ- entiating term-by-term the power series that defines e z . 8. From 238, 1, we also deduce the special values III. The functions cos z and sin z. In the case of the trigonometrical functions, we can again use the expansions in power series convergent everywhere to define the functions 239* for complex values of the variable. Definition. The sum of the power series, convergent everywhere, z 2 z* z 2k 1 "" 2 ! + il - + + (- *)* (2 k)\ + ' ' ' ' is denoted by cos z, that of the power series, also convergent everywhere^ 1~! ^ I ' FT I r + ( 1, 1 ! O ! J ! by sin z, for every complex z. 55. The elementary analytic functions. III. The functions cos z and sin z. 415 For real z = x, this certainly gives us the former functions cos x and sin x. We have only to verify, as before, whether these defini- tions are suitable ones, in the sense that the functions defined, which are analytic in the whole plane, i.e. integral functions, possess the same fundamental properties as the real functions 30 cos x and sin x. Thai this is again the case, to the fullest extent, is shown by the following statement of their main properties: 1. For every complex z, we have the formulae 240. cos #+ /sin # = 0*% whence further e ** + e -** . - -~ - , 81112 (Eider's formulae). The proof follows immediately by replacing the functions on both sides by the power series which define them. 2. The addition theorems remain valid for complex values of z: cos (z l + 2 3 ) = cos z t cos z.} sin z 1 sin z^, sin (^ -f- 2 2 ) = cos 2 i sm ^3 + sm %i cos -? 3 f This follows from 1., since by 237 and the latter involves cosfo +* 9 ) + isin(* 1 +* 9 ) == (cos 2j + i sin zj (cos z a -f- i sin z a ) = (cos Z A cos z 2 sin ^ sin ^) -f- i (cos ^ sin z 2 -f sin 2:, cos *,) . Substituting z and z^ for ^j and 2 a , and taking into account the fact that cos z is an even, sinz an odd function, we obtain a similar formula, which differs from the last only in that i appears to be changed to i on either side. Addition and subtraction of the two relations give us the required addition formulae. 3. The fact that the addition theorems for our two integral func- tions are formally the same as those for the functions cos a; and sin a; of the real variable x, not only sufficiently justifies our designating these functions by cos,? and sins, but shows, at the same time, that the entire formal machinery of the so-called goniometry t since it is evolved from the addition theorems, remains unaltered. In particular, 90 Here again a remark analogous to that on p. 412, footnote 28, may be made. 416 Chapter XII. Series ot complex terms. we have the formulae cos 3 z -f- sin 2 z = 1, cos 2 z = cos 9 z sin z 9 sin 2 z = 2 sin z cos z, etc. valid without change for every complex z. 4. The period-properties of the functions are also retained in the complex domain. For it follows from the addition theorems that cos (z -f- 2 n) = cos z cos 2 n sin z sin 2 rc = cos 2, sin (2 -f- 2 JT) = cos 2 sin 2 JT + sin 2 cos 2 JT = sin 2 . 5. 77*0 functions cos z aw<2 sin z possess no other zeros in the com- plex domain besides those already known in the real domain* 1 . In fact, cos z = necessarily involves, by 1., e iz = e~ iz or i. e. By 238, 4, this can only occur when Similarly, sinz = implies e iz ~ e" iz , or e* iz = 1, i. e. 2 iz = 2 k jit, or z = kn, q. e. d. 6. The relation cos z = cos z 2 is satisfied if, and only if, z ^ = -j- z ^ -. 2 kn, i. e. under the same condition as in the real domain. Similarly sm z^ = sin z^ if, and only if t z^ = z -f- 2 k n or z g = n z + 2 k n. It follows in fact from cos z, cos 2 2 = 2 sm -~^ - 2 sin - 1 -,, 3 - = , > u by 5., that either - JL -Q fl r S>~ 2 must = ft JT; similarly it follows from i & sin z t sin z% = 2 cos L g g sin -^-Q--- = , by 5., that either -^^ = ft ^ or ^-^ = (2 A + 1) -*- . 7. 77z functions cos,? aw^ sin^r assume every complex value w in the period-strip, i. e. in the strip n < $1 (z) <I -f- n, the equations cos 2 = w and sin 2 = w have indeed exactly two solutions in that strip, if w ^ l, but only one, if ze> = 1. 81 Or in other words: The sum of the power series 1 - -| ... = if, and only if, z has one of the values (2 k -f 1) , k = 0, 1 , 2, ...; and A .similarly for the sine series. g 55. the elementary analytic functions. iV. ; lhe functions cot z and tan z. Proof. In order to have cos z ~- w, we must have e iz -\- e~ iz 2 10 or e iz = w -f- Vw 2 1. (Here V r (cos r * = w -f- V w* 1. (Here V Y (cos r/; + i sin g?) is defined as one of the two numbers, for instance 72(0)8-?* +isin-~V whose square is the quantity under the radical c ign.) Since in any case 32 w + Vw 2 1 =}= 0, there certainly exists a complex num- ber / such that - n < 3(Y) < +n, foi which e z ' = w -\-V w* 1, by 238,5. Writing i z ' = z, we have n < 9R (2) <^ = w + ' Fig. 12. ji and 12 or cos z = w. This equation therefore certainly has at least one solution in the penod-stnp. By 6., however, a second solution, different from it, (viz. z), exists in the period -strip if, and only if, z =H and =f= n, i. e. w + 1- We reason in precisely the same manner with regard to the equation sin z -- w. In this case, we can also easily convince our- selves that there is always one and only one solution of the equation in the portion of the period-strip left unshaded in Fig 12, if we in- clude the parts of the rim indicated in black, but omit the parts re- presented by the dotted lines (see VI below). 8. For the derivatives, we have as in the real case, (cos z)' sin 3, (sin z)' cos z. IV. The functions cot# and tan^gr. 1. Since cos z and sin z are analytic in the whole plane, the functions cot z = ^? and sin z tan z = -- cos z will also be regular in the whole plane, with the exception of the points k TT for the former and (2 k + 1) for the latter, which are the zeros of sin z and cos z respectively. Their expansions in power series may be obtained by carrying out the division of the cosine and sine series. Since this operation is of a purely formal nature, the result must be the same as it was in the real domain. Accordingly, by 24, 4, where the result of this division was obtained by a special artifice, we have k - t n __ / 1 \fc-l tan ~ -Z (- 1) 32 In fact, since w* 1 =+= w 8 , Vw 2 1 4= i ?. 418 Chapter XII. Series oi complex terms. On account of 94 and 136, we are now also in a position to de termine the exact radius of convergence of these series. The absolute value of the coefficient of z 2k in the first series, by 136, is Its (2A) th root is if So t denotes the sum 5? ^r. The latter lies between 1 and 2 for &K *^ ft VK every k = 1, 2, ... (for it is -^ when k = 1, and is less than this for every other i, but > 1); therefore 1 -*' 12*) i "^ IT' and the radius of the cot-series = n, by 94. Similarly that of the tan-series is found to be -^ . 2. cot z and tan* Aatte tf/te period n. For cos 3 and sin^r ftott change in sign alone when z is increased by n. Here again we may show, more precisely, that cot * a = cot jgr a and tan jgr a = tan jgr a involve *, = *, + ** (*-0, 1,...). In fact, it follows from cos z. cos z 9 sin fo, *.) COt Z, COt &. = : : = : . X 1 2 sin 2r t sin 2 2 sin ^ sin z^ that in the case of the first equation sin (z a zj == 0, i. e. z a z = kn. Similarly in the case of the second. 3. In the "period-strip", i. e. in the strip ^ < SR (0) <I + ~ , cot 2r and tan 2 asswme every complex value w =f= * /ws^ once; /Ae values wi are never assumed. To see this, write 2 *==. The equation cot 2 = 10 then becomes For each w *%* i, is a definite complex number 4 s an( * (by II, 5) there accordingly exists a / such that n <. % (/) <^ n> for which e* = f . For z = *'-^, we then have u ~ < SR (*) <; y and cot jar = w, i. e. z is a solution of the latter equation in the prescribed strip. By 2. there can be no other solution in this strip. The impossibility g 55. The elementary analytic functions. V. Ihe logarithmic series. 419 of a solution for cot z = * results from the fact that these equations both involve which cannot be satisfied by any value of z, as cos a z + sin a = 1. For tans the procedure is quite similar. 4. The expansion in partial fractions deduced in 24, 5 for the cotangent in the real domain remains valid in the same form for every complex z different from 0, 1* it 2,... (and similarly for the ex- pansions of tans, etc.). Indeed the complete reasoning given there may be interpreted in the "complex" sense, without altering a single word 33 . In particular, for every z satisfying the above con- dition, Now it follows, if we substitute z for 2inz, that _ , 2 ^_ hence we obtain the expansion i i 1 valid for every complex z=$=2kni (&0, integer). This is the ex- tension to the complex variable z of the remarkable expansion in partial fractions obtained on p. 378, and it exhibits the true connection between this expansion and that of cots, which previously seemed rather fortuitous. V. The logarithmic series. In 25, we saw that the series represents for every | x \ < 1 the inverse function of the exponential function e v 1; i. e. substituting for y in 3! 33 It was precisely for this purpose that at the time we framed some of our estimates in a form somewhat different from that required for the real domain (e. g. those on pp. 200 207 to which footnote 26 refers). 420 Chapter XII. Series of complex terms. the above series and rearranging (as is certainly allowed) in powers of x, we reduce the new series simply to x. This fact because it is purely formal in character necessarily remains when complex quantities are considered. Hence, for every | z \ < 1, e" 1 = z or e w = 1 + z, if w denotes the sum of the series if\ T (l) 71 " 1 v n {L.) W A Z . We now adopt for the complex domain the 242. Definition. A number a is said to be a natural logarithm of c, in symbols, a =- log c f if e = c. In accordance with II, 5, we may then assert that every complex number c 4= possesses one, and only one, logarithm whose imaginary part lies between TT exclusive and + ^ inclusive (to the number 0, however, by II, 6, no logarithm can be assigned at all). This uniquely defined value will be more especially referred to as the principal value of the natural logarithm of c. Besides this value, there is an infinity of other logarithms of c, since with e a = c we have also e a + 2k = c\ thus if a is the principal value of the logarithm of c, the numbers a + 2 k TT i (k ^ 0, integer) must also be called logarithms of c. These values of the logarithm (for k =*= 0) are called its subsidiary values M . By 238, 4 there can be no further logarithms of c. We have, for each of its values, ffl (log*) = log | c | , 3 (log c) = amc, if in the first of these relations log | c \ denotes the (single-valued) real logarithm of the positive number \c\, and the second is interpreted as meaning that, taken as a whole, all the values of the one side are equal to all the values of the other. With these definitions, we may assert in any case that the above series (L) provides a logarithm of (1 + #) But we mav at once prove more, namely the 43. Theorem. The logarithmic series (L) gives, at each point of the unit circle (including its rim, with the exception of the point 1), the principal value of log (1 + #). 34 If c is real and positive, the principal value of log c coincides with the (real) natural logarithm as formerly defined (36, Def.). 55. The elementary analytic functions VI. The inverse sine series. 421 Proof. That the series converges for each z =t= - 1 for which | z | ^ 1 was shown in 230, 3. (We have only to put % for z there.) For this z, am(l -|- #) has precisely that value iff for which 7T / -. I TT 2 < * < + 2- Hence we have, for the imaginary part of the sum w of the series (L), (3) 3(w) = with integral k. Now w is a continuous function of z in | z \ <. 1, and assumes the value 1 for xr 0. Hence 3 (w) too is a continuous function in | # | < 1. Therefore, in the equation (^)> ^ must have the same value for all these z. But for JST \\e have clearly to take k 0; hence this is its value in the whole of j % \ < 1. Finally \ve learn from the application of Abel's limit theorem that the sum of our series is still equal to the prin- cipal value of log (1 + z) at the points 2 =|= 1 for which | z \ = 1. VI. The inverse sine series. We saw in III, 7 that the equation sin w = #, for a given complex ~ ^ 1, has exactly two solutions, for xr - -[ : 1 exactly one, in the strip TT < j)f (77) ^ + TT. The two solutions (by III, 6) arc sym- metrical, either with respect to + \ or ^; accordingly, we may assert more precisely that the equation sin w ^- z, for an arbitrary given z (in- clusive of 1), has one and only one solution in the strip - I ^ SB (w) ^ + f , if the lower portions of its rim, from the real axis downwards, arc omitted (cf. Fig. 12, where the parts of the rim not counted with the strip are drawn in dotted lines, and the others are marked by a continuous black line). This value of the solution of the equation sin w = z, which is thus uniquely defined for every complex ^r, is called the principal value of the function tv sin" 1 z. All the remaining values arc contained, by III, 6, in the two formulae sin" 1 z -\- 2k TT, TT sin" 1 z + 2 k TT, and may be called subsidiary values of the function. 422 Chapter XII. Series of complex terms. For real values of x such that |as| <l, the series 123, . 1 x 9 , 1-8 * B , ? ==a; + -2--3- + 274Tr + - represents the inverse series of the sine power series Exactly the same considerations as in V. for the case of the log- arithmic series now show that, for complex values of z such that | z \ <J 1 , the series . 1 z 9 . 1-3 * B , "' + T-T+2-4-5- + "- w* is the inverse series of the sine power series w -- ^7 -| -- . It therefore gives at any rate one of the values of sin" 1 z. That this actually is the principal value, may be seen from the fact that, for == sin" 1 |z| <: sin"" 1 1 = -J, a condition which the principal value alone fulfils. VII. The inverse tangent series. The equation tant0 = z, as we know from IV, 3, has for every given z =|= i i one and only one solution in the strip -- - < 91 (w) <^ -f" -^- . This is called the principal value of the function the other values of which (by IV, 2) are then obtained from the formula tan" 1 z + k 7t . The equations tan z = i have no solutions whatever. Almost word for word the same considerations as above again show that, for | z \ <T 1 , the series (A) = 2 _.+._ +... gives one of the solutions of tznw z. To show that this is actually the principal value of tan"" 1 ,?, we have to show that the real part of the sum of the series lies between -- 5- (exclusive) and -f- -5- (inclusive). This remains true for every z 4 s i on | * | =* 1> as we ^ as for | ar | < 1, and is proved as follows: The sum w of the series (A), as may be seen by substituting the log-series, is w = -^j-log (1 + iz) -^-log (1 iz) 55. The elementary analytic functions. VIII. The binomial series. 423 for every \z\ <[ 1, z 4 s i> where principal values are taken for both logarithms. Accordingly, SRW = y3log(l + ^)~|3lo g (l-^); by 243, both terms of the difference lie between - and + ~ , hence fR (w) lies between ~ and -f- ~rr > me two extreme values being excluded in either case. Thus the series (A) certainly represents the principal value of tan" 1 ,?, provided |z|<^l and z + i> q- e. d. VIII. The binomial series. To complete our present treatment of the special power series in- vestigated in the real domain, we have only to consider the bino- mial series in the case where the quantities occurring there i. e. the exponent a as well as the variable x assume complex values. We start with the Definition. The name of principal value of the power b a , where 244. a and b denote any complex numbers, with b 4 s as the only condi- tion, is given to the number uniquely defined by the formula when log b is given its principal value. By choosing other values of log b 9 we obtain further \alues of the power, which may be called its subsidiary values. All these values are contained in the formula ta ^a[\ogb+2kai] u 6 , each value being represented exactly once, if log b is given its prin- cipal value and k takes all integral values ^0. Remarks and Examples. 1. A power 6 fl accordingly has an infinite number of values in general, but possesses one and only one principal value. 2. The symbol **, for instance, denotes the infinity of numbers (all real numbers, moreover) , ( ft= , 1. 2, .. ) a of which e 2 is the principal value of the power **. 3. The only case in which a power b a will not have an infinite number of values is that in which *" (ft-0, 1,2. ...) 424 Chapter XII. Series of complex terms. gives only a finite number of values; this will occur if, and only if, ka assumes, for = 0, 1, 2, ..., only a finite number of essentially different values. Here two numbers arc described (just for the moment) ,is essentially different if, and only if, they do not differ merely by a (real) integer. i\ow this is the case if, and only if, a is a real rational number, as may be seen at once; and the number of "essentially different" values which may in this case be assumed by k-a is given by the smallest positive denominator with which a may be written in fractional form. J. 4. Tt follows that b m = yb, where m is a positive integer, has exactly m different values, one of which is quite definitely distinguished as the prin- cipal value. ">. Tne number of different values of b a will reduce to one, by 3. and 4., if, and only if, a is a rational number of denominator I, i. e. a real integer. For all real integral exponents (but for these alone), the power thus remains now as before a single-valued symbol. 6. If b is positive and a real, the value formerly defined (v. 33) as the power b a is now the principal value of this power. 7. Similarly, the values defined in 23G for e z and p z , (>0), are now, more precisely, the principal values of these powers In themselves, these sym- bols would represent, for complex values of z, an infinity of vakus, in ac- cordance with our last definition. Nevertheless, we shall keep in future to the convention that e z , and generally p z for any positive p t shall represent the value defined by 236, i e the principal value only 8. The following theorems will show that it is consistent to define b a also for b = when fll(a);>0. The value attributed to the po\\er in th.it case is (uniquely). After making these preliminary preparations, we proceed to prove the following far-reaching Theorem 35 . For any complex exponent a and any complex z in | z | < 1 , the binomial series converges and has for sum the principal value of the power (1+1)-. Proof The convergence follows word for word as in the case of real zs and 's (v pp 200 -210), so that we haxr only to prove the statement as to the sum of the series. Now for real x's such that \x \ < 1, and real a's, we may substitute T. f d remr n anjrrw Math, V 1, p. 311 1826. 55. The elementary analytic functions. VJII. The binomial series. 425 V" for y in the exponential series = ! + )/ + --] and so obtain, after learranging in powers of x fallowed by 1O4), the power series for s log u + *) = (l -f ar), i. e. the binomial series ("\x n . Let us proceed in this manner, purely formally in the first instance, assuming complex and writing z for x; i. e. we substitute 'x* ( ]\ n 1 ^ iti n w = a .2 { -^ z n in e^^Z^-r nl n n=l nl and rearrange in powers of z. We necessarily obtain without refer- ence as yet to any question of convergence the series whose sum would therefore be proved to be gi (*+*) = (l -f- z) a (where the principal value is taken for the logarithm and hence for the power also), if we could show that the rearrangement carried out was per- missible. Now by 1O4 this is certainly so; in fact the exponential series converges everywhere and the series cc - -- z n remains convergent for | z | <. 1 when a and all the terms of the senes are replaced by their absolute values. This proves the theorem in its full extent. If we split up (1 + z) a into its real and imaginary parts, we obtain a formula due to Abel, which is complicated in appearance, but which for that very reason shows how far-reac hing a result is contained in the preceding theorem, and from which we also obtain a means for evaluating the power (1 -f- z) a . Writing z = r (cos cp -f- i sin cp) and a = /3 + iy, < r < 1 , r/>, /?, / all real, and writing 1 + z = R (cos + i sin 0) , we have R = Vl + 2ycos//? + y' 2 , = principal value 3G of tan- 1 - With these values of R and 0, we thus obtain = Rfi . e -y * [cos (00 + y log R) + i sin (ji& + y log R)] . For the case |^|<1, theorem 245 and the remark just made completely answer the question as to the sum of the binomial series. We have now only to consider the points of the circumference | z \ = 1 . From Abel's theorem, together with the continuity of the principal value of log (1 -f- z) for every 2 =}= 1 in | z \ <[ 1 and the continuity of the exponential function, we at once deduce the has accordingly to be chosen between + -5- and -H~- 426 Chapter XII. Series of complex terms. 246. Theorem. At every point of the rim |2| = 1 of the unit circle, at which the binomial series continues to converge, except possibly for 2 = 1, its sum remains now as previously the principal value of (! + *)-. The determination whether, and for what values of a and z, the binomial series continues to converge on ike rim of the unit circle presents no difficulties after the preparations made in this respect (and chiefly for this purpose) in 53. The theorem we have is the following, which sums up the entire question once more: 00 i 247. Theorem. The binomial series J^( }z n reduces, for real integ- n=o > n ' ral values of tf^O, to a finite sum, and has then the (ipso facto unique) value (\-\-z) a \ in particular for a = it has the value 1 (also when z = 1) . // # does not have one of these values, the series con- verges absolutely for \ z \ < 1 and diverges for \z\ > 1 , while it exhibits the following behaviour on the circumference \ z \ = 1 : a) if 91 (a) > , it converges absolutely at all points on the circum- ference', b) if 91 (a) ^ 1, it diverges at all these points; c) if 1 < 91 (a) <I 0, it diverges at z = 1 and converges con- ditionally at every other point of the circumference. The sum of the series when it converges is invariably the principal value of (I -\- z) a ; in particular, its value is in the case z = 1. Proof. Writing ( l) w (") = n + 1 , we have *-(+!) f a \ U-i/ . __ hence theorem 229 may be applied, and the validity of a), b) and c) follows immediately. Only the case of the point z = 1, i. e. the con- vergence of the series requires special investigation. Now a (a - 1) (a - 2) 55. The elementary analytic functions. IV. The binomial series. 427 and in general, as may at once be verified by induction: the partial sums of our series are equal to the paitial products, with 00 / cc \ the same index n, of the product //fl ) The behaviour of this product is immediately evident. In fact 1. If <K(a) = /?>0, choose ft* such that </?'</?; for every sufficiently large n, say n^tm, hence By 12t>, 2, it follows at once that the partial products, and hence the partial sums of our series, tend to 0. The series therefore converges 37 to the sum 0. 2. If, however, R (a) = ft < 0, we have a n whence it again follows by multiplication that and hence that the left hand side tends to oo. The series therefore diverges in this case. 3. If, finally, $ (a) ~ 0, a = i y, say, with y ^ 0, the w th partial sum of our scries is The fact that this value tends to no limit as n + + ma y De proved most speedily in the present connection as follows: On account of the ab- solute convergence of the series J?( J , we have, by 29, theorem 10. Letting w> + oo, the right hand side evidently tends to no limit; on the contrary, the points which it represents for successive values of n circulate incessantly round the circumference of the unit circle in a constant sense, the interval between successive points becoming smaller 37 The mere convergence of (!)*( J follows already from 228 and we see that the convergence is absolute when SR(a)>0. It is the fact of the sum being which requires the artifice employed above for its detection. 428 Chapter XII. Series of complex terms. and smaller at each turn. In view of the asymptotic relationship, the same is therefore true of the left hand side. Hence our series J( l) w ( J also diverges when s Ji(a) = 0. Thus theorem 247 is established in all its parts, the behaviour of the binomial series is de- termined for every value of z and of a, and its sum for all points of convergence is given by means of a "closed expression". 56. Series of variable terms. Uniform convergence. theorem on double series. The fundamental remarks on series of variable terms n=0 are substantially the same for the complex as for the real domain (v. 46); but instead of the common interval of definition we must now assume a common region of definition, which for simplicity this is also quite sufficient for most purposes we shall suppose to be a circle (cf. p. 403, footnote 17). We accordingly assume that 1. A circle \z z \ < r exists, in which the functions f n (z) are all defined. 2. For every individual z in the circle \ z Z Q \ < r, the series n=0 is convergent. The scries 2 f n (z) then has, for every z in the circle, a definite sum, whose value therefore defines a function of z (in the sense of the definition on p. 403). We accordingly write 00 2fn(*) = F(z). =0 The same problems as those discussed in 46 and 47 for the case of real variables anse in connection with the functions represented by complex series of variable terms. In the real domain, however, it is of the greatest importance, both for the theory and its appli- cations, to make use of the concept of function in its most general form, while in the complex domain this has not been fou:id profitable. The usual restriction, which is sufficiently wide for all ordinary pur- poses, is to consider analytic functions only. We therefore assume further that 3. The functions f n (z) are all analytic in the circle \ z 2 | < r, i. e. expressible by power series with z as centre and radius not less than a fixed number r. $ 56. Series of variable terms. 429 We then speak for brevity of series of analytic functions**; the chief problem concerning such a series is the following: Is the function F(z) which it represents analytic in the circle \z | < ?> or not? Precisely as in the real domain, it may be shown by examples that without further assumptions this need not be the case. On the other hand, the desired behaviour of F(z) may be ensured by stipul- ating (cf. 47, first paragraph) that the series converges uniformly. The definition for this is almost word for word a repetition of 191: Definition (2 nd form 39 ). A series 2 f n (z), all of whose terms are 248. defined in the circle \ z z a \ < r or in the circle \ z Z Q \ <^ r, and which converges in this circle, is said to converge uniformly in this circle if, for every e > 0, it is possible to choose a single number N > (independent, therefore, of z) such that for every n > N and every z in the circle considered. Remarks. 1. Uniformity of convergence is here considered relative to all the points of an open or closed circle* . Of course other types of region or indeed arcs of curves or any other set $)t of points, not merely finite in number, may be taken as a basis for the definition. The definition remains the same in sub- stance. Tn applications, we shall usually be concerned with the case in which the terms f n (z) are defined, and the series 2f n (z) converges, at every point interior to a circle |* z |<; r (or a domain 0)), but the convergence is uniform only in a smaller circle \ z Z Q \ < g, uhcre < r, (or in a smaller sub- domain j, which, together with, its boundary, belongs to the interior of (i)) 2 If the power series ~a n (z z ) n has the radius r, and 0<0<>, the series is uniformly convergent in the (closed) circle \Z ZQ\<Q- Proof word for word as on p. 333. 3. If r is the exact radius of convergence of 2 a n (z z ) n , the conver- gence is not necessarily uniform in the circle | z z \ < f . Example the geo- metric series, proof on p 333. 4. Exactly as befoie, we may verify that our definition is completely equivalent to the following: 88 Here again we may remark (cf. 190, 4) that there is no substantial difference between the treatment of series of variable terms and that of sequences of functions A series 2f n (z) is equivalent to the sequence of its partial sums s (z), Sj (z), . . ., and a sequence of functions s n (z) is equivalent to the series s fl (*)+(, (*) -* (*))H For simplicity, we shall hereafter formulate all definitions and theorems for series alone; the student will easily be able to enunciate them for sequences. 89 This definition corresponds to the former 2 nd form. The 1 st form 191 may here be omitted, as it did not appear essential for the application of the concept of uniform convergence, but only for its introduction. 43 The set of points of a circle (or, for short, the citcle itself) is said to be closed or open according as the points of the circumference are regarded as included in the set or not. 430 Chapter XII. Series of complex terms. 3 rd form. S f n (z) is said to be uniformly convergent in \ * * | < Q (or in the set 9ft), if, for every choice of points z n belonging to this circle (or set), the corresponding remainders r n (z n ) always form a null sequence. The 4 th and 5 th forms of the definition (p. 335) also remain entirely un altered and we may dispense with a special statement of them here. On the other hand, it is impossible to give as impressive a geometrical representation of uniform and non-uniform convergence of a series as in the real domain. We are now in a position to formulate and prove the theorem announced. 249. Weierstrass 9 theorem on double series 41 . We suppose given a series each of whose terms f k (z) is analytic at least for \z z \ < r, so that the expansions 42 all exist and converge at least for \z z Q \<r. Further, we assume that the series 2f^(z) converges uniformly in the circle \ z z \ < , for every Q < r, so that the series converges, in particular, everywhere within the circle \z z Q \<r, and represents a definite function F(z) there. It may then be shown that. 1. The coefficients in a vertical column form a convergent series: , = 0, 1,2,...)- 2. ^j A n (z z ) n converges for \ z z \ < r . n=0 3. For | z Z Q | < r, the function is again analytic, with n 41 Werke, Vol. 1, p. 70. The proof dates from the year 1841. 42 The upper index, in the coefficient a n M, indicates the place occupied in the given series by the corresponding function, while the lower index relatec to the position, in the expansion of this function, of the term to which th ?o 56. Series of variable terms. 431 4. For | z z | < r and for every (fixed) v 1, 2, . . . , k-o i. e. the successive derived functions of F (z) may be obtained by term-by-term differentiation of the given series , and each of the new series converges uniformly in every circle \ % # | 5^ , with g < r. Remarks. 1. If we direct our attention primarily to expansions in power series, the theorem simply states that with the assumptions detailed above, an infinite number of power series "may" be added term by term. If on the other hand we look rather at the analytic character of the various functions, we have the following Theorem. If each of the functions f k (z) is regular for \ z s | < r and the series 27/ fc (z) converges uniformly in \ z z \ ^ g, for every g < r, then this series represents an analytic function F (z), regular in the circle \ z Z Q \ < r. The succes- sive derived functions JFX") (z) of F (z) t for every v ^ 1, are represented^ in that circle \ by the serie* </&(") (z), obtained from 2f k (z) by differentiating term by term, v times in succession. Each of these series converges uniformly in every circle \ z ar | 5^ g, with Q < r. 2. The assumption that 27/ 7c (z) converges in | z # | 5C g for every g < r is satisfied, for instance, by every power series 27 c k (z z Q ) k with radius of con- vergence r. It is also satisfied e. g. by the series J 1 _ - -, for r = 1 ; cf. 58, C. *^ 1 ~~~ s 3. The first of our four statements shows that the present theorem cannot be proved simply as an application of Markoff's transformation of series; for the latter assumes the convergence of the columns, here this is deduced from the other hypotheses. Proof. 1 . Let an index m, a positive g < r and an e > be chosen to be kept fixed throughout. By hypothesis, we can determine a k Q such that, throughout | z Z Q \ <^ g, for every k such that k' > k > A , if we write ** = ** (*) =/o (*) + ...+/* (*) Now the function s k > (z) s k (z) is a definite power series, whose i7t th coeiBcient is By Cauchy's inequality 235, we therefore have Hence the series (a) *W +<> + + + = A-0 is convergent, by 81. Let A m be its sum. As m could be chosen arbitrarily, the first of our statements is thus established. 432 Chapter XII. Series of complex terms. 2. Now let M' be the maximum 43 of | s fco ^ i (z) \ along the circum- ference | z - Z Q | = y. We have then for every k > k on the same cir- cumference I ** (*) I ^ I **.+ 1 (*) I + I ** (*) - % + i (*) \M' + t' = M. Again, using Cauchy's inequality, we obtain, for every n - 0, 1, 2, . . . , whatever the value of k. Hence M 00 and 27 A n (z z Q ) n therefore converges for | z z \ < &. Since the only n^O restriction on g was that it should be < r, the series must even converge for | z z | < r. (In fact, if z is any determinate point satisfying the inequality | z Z Q \ < r, it is always possible to assume Q to be chosen so that | z Z Q | < Q < r.) Let us for the moment denote by F l (z) the function represented by the series A n (z #o) n " ft ^ s thus, by its defini- tion, an analytic function in | z ^ | < r. 3. We have now to show that F l (z) = F (z), so that F (z) is itself an analytic function regular in | z Z Q \ < r. For this purpose, we choose, as in the first part of our proof, a positive (/ < r y a positive Q in Q' < # < r, and an e > 0, fixed. We can determine k so that, for all z in | z xr | fj , for every k such that k' > k > & . By Cauchy's inequality, it follows as before that, for k' > k > k and for every n ^ 0, Making &' -> + , we infer that, for every k > k and every n S 0, I A n - ( + <*> + + an*) I ^ Now the expression between the modulus signs is the w th coefficient in k the expansion of F l (z) Zf v (z) in powers of (z z ). Hence we have, i>=o for | z Z Q | < g: | F, (z) - If/, (*) \*' v=-0 The right hand side is, for | z z \ <^ Q', 43 I S k +i ( z ) I is a continuous function of am z 9 along the circumference in question and (9 being real) attains a definite maximum on this circumference. 56. Series of variable terms. 433 Thus, when e > and Q' < Q < r have been chosen arbitrarily, we can determine k so that for every k > k Q and every \z Z Q \ <^ (/. This implies, however, that for these values of z ^ r=0 The numbers @' and were subjected to no restriction other than < Q' < Q < v\ hence (as above) it follows that the equation holds for every z interior to the circle | z z \ < r. 4. We write /o' (*) = V 1 " + 2 ./ 0) (z - z ) + 3 a a (z - z ) 9 + fi' () ^ i (1) + 2 s (1) (* - *o) + 3 V (* - *o) 9 + ' ' ' A! + 2 AS (z- z ) + iA s (z - 2 ) 2 + . ., where the sum of the coefficients in any one column converges to the value written immediately below them. Just as in 3. (we have only to begin our evaluations with e' = ( Q Q Y e) we deduce that for | z Z Q \ ^ e r < Q < r and every k > 7e , Hence for those values of z y F' (z) = 2f k ' (z). Indeed, by the same A-O* reasoning as before, this series converges uniformly in | z Z Q \ < r, for every g <. r. If we write down the corresponding system of series for the v th derived functions, we obtain, in the same manner: F<"> (z) = 27/fcW (z) (v = 1, 2, . . . , fixed) k o for every | z # |<:r; i.e. the scries Zf^(z) obtained by differ- entiating term by term, v times in succession, converges in the whole circle | z Z Q \ <r (and converges uniformly in every circle | z z \ 5^ Q < r) and gives the i> th derived function of F (z) there. Remarks. 1. A few examples of particular importance will be discussed in detail in the next section but one. 2. The fact of assuming the convergence uniform in a circular domain is immaterial for the most essential part of the theorem: If G is a domain of arbitrary shape 44 and if every point Z Q of the domain is the centre of a circle | z ar | g g (for some Q) which belongs entirely to the domain, is such that each term of the series Zfj. (z) is analytic there, and is a circle of uniform convergence of the given series, then this series also represents a function F (sr) analytic in the domain in question, whose derived functions may be obtained by differentiation term by term. Examples of this will also be given in 58. " Cf. p. 403, footnote 17. 434 Chapter XI 1. Series of complex terms. 57. Products with complex terms. The developments of Chapter VII were conducted in such a way that all definitions and theorems relating to products with "arbitrary" terms hold without alteration when we admit complex values for the factors. In particular the definition of convergence 125 and the theo- rems 1 , 2 and 5 connected with it, as well as the proofs of the latter, remain entirely unchanged. There is also nothing to modify in 127, the definition of absolute convergence, and the related theorems 6 and 7. On the other hand, some doubt might arise as to the literal trans- ference of theorem 8 to the complex domain. Here again, however, everything may be interpreted as "complex", provided we agree to take log (1 + a J to mean the principal value of the logarithm, for every sufficiently large n. The reasoning requires care, and we shall therefore carry out the proof in full: 250. Theorem. The product 77(1 + 0J converges if, and only if, the series, starting with a suitable index m, whose terms are the principal values of log (1 + a n ), converges. If L m is the sum of this series, we have, moreover, 77(1 -|- a n ) = (1 + a,) (1 + * 2 ) . . . (1 + a m ) *'-. n-i Proof, a) The conditions are sufficient. For if the series O w * m tne principal values of the logarithms, is con- n-m+i vergent, its partial sums s n , (n > m), tend to a definite limit L, and consequently, since the exponential function is continous at every point, i. e. it certainly tends to a value + 0. Hence the product is con- vergent in accordance with the definition 125 and has the value stated. b) The conditions are necessary. For, if the product converges, given a positive e, which we may assume < 1, we can determine n so that (a) | (1 for every n ^> n Q and every fcj>l. We then have, in particular, I a n | < ~ < -g- for every n > n , and the inequality | a n \ < -- is thus certainly fulfilled for every n greater than a certain index m. We may now show further that for the same values of n and k hisine' the 67. Products with complex terms. 435 principal values of the logarithms) 45 (b) n I k E \ <e and therefore the series 27 log (1 + a n ) is convergent. In fact, as | a v \ < ^ Ji-mi-l * for every v > 0> we also have 46 , for these values of v, (c) I log (! + ,) | <e, and likewise, by (a), I log [(1 + re+1 ) . . . (1 + +*)] I < * for every w ^ w and every k ^ 1. Accordingly, for some suitable integer 47 q y we certainly have | log (1 + a n+ J + log(l + a n ^) + . . . + log(l + a n+k ) + 2qni\<s, and it only remains to show that q may in every case be taken = 0. Now if we take any particular n ^ # , this is certainly true for k = 1, by (c). It follows that it is true for k = 2. For in the expression log (1 + a n+1 ) + log (1 + a n+2 ) + 2q*i the modulus of either of the two first terms < e, by (c), and by (d) the modulus of the whole expression has to be < e; as e < 1, q cannot, there- fore, be an integer different from 0. For corresponding reasons, it also follows that for k = 3 the integer q must be 0, and this is then easily seen by induction to be true for every k. This establishes the theorem. The part of theorem 127, 8 relating to absolute convergence may also be immediately transferred to the complex domain, viz. no 00 the series 2 log (1 + a n ) and the product // (1 + a n ) n = m + 1 n -m \ \ are simultaneously absolutely or non-absolutely convergent, in every case. Similarly the theorems 9 11 of 29 and 30 remain valid. In fact, it remains true for complex a n y s of modulus < - that in 45 The logarithms are always taken to have their principal values in what follows. 46 In fact, for | a | < ~, I log (i + *) | ^ | * | + L* |a + . . . ^ | * ! + i * p + . . . = -jJ* Lj < 2 1 z | . 47 For the principal value of the logarithm of a product is not necessarily the sum of the principal values of the logarithms of the factors, but may differ from this sum by a multiple of 2 TT i. Thus e. g. log i = k> - , but log (i *"" = log 1 = 0, if we take principal values throughout. 436 Chapter Xll. Series ot complex terms. the quantities ?^ n are bounded, since when | z \ < 4 . while the expression in square brackets clearly has its modulus < 1 for those z's. Finally, the remarks on the general connection between series and products also hold without alteration, since they were purely formal in character. 251* Examples. * IL (1 + "-) is Divergent. For 2 \ a n | 2 == s is convergent, so that by 29, theorem 10, the partial products the right hand expression represents, for successive values of , points on the circumference of the unit circle, which circulate incessantly round this circum- ference at shorter and shorter intervals. p n therefore tends to no limiting value. (Cf. pp. 4278.) 2. 21 f n n~"'i =s '" 1 ' In fact > the wth P artial Product is at once 8. For | z |< 1, 7/ (1 -f- * 2 ") = :p . In fact the absolute) convergence of this product is obvious by 127, 7 and its n th partial product multiplied by (1-*) is which tends to 1. The consideration of products whose terms are functions of a complex variable, n=l like that of series of variable terms in the preceding section, will be restricted to the simplest, but also the most important case, in which the functions f n (z) are all analytic in one and the same circle | z _ ZQ | <; r (i. e . possess an expansion in power series com crgent in that circle) and in which the product also converges everywhere in the circle. The product then represents a definite function F(z) in the circle, which is said, conversely, to be expanded in the given product. We next enquire under what convenient conditions the function F(z) represented by ihe product is also analytic in the circle |2 2 |<?. For the great majority of applications, the following theorem is sufficient: 57. Products with complex terms. 437 Theorem. // the functions f l (xr), / 2 (xr), . . . , f n (xr), . . . are all analytic 252. at least in the (fixed) circle \ z xr | < r ; if, further, the series n-l converges uniformly in the smaller circle \ z z \ ^ g, for every positive (j < r\ then the product //(I +/ n (#)) converges everywhere in \ z xr | < r and represents a function F (z) which is itself analytic in that circle. The proof follows the same line of argument as that of the continuity theorem 218, 1 almost word for word. To establish the con- vergence and analytic character of the product at a particular point z l in the circle | z z | < r, we choose a g < r and prove the two facts first for every % of the circle | z Z Q \ <Q. The series 2 \ f n (z) \ converges uniformly in the whole of | z # | ^ g, so that the product //(I +/ n (z)) certainly converges there (indeed absolutely). Choose m so large that verges there (indeed absolutely). Choose I / m +l (*) I + I /,+ 2 (*) I + + I fn (*) | < 1 for every n > m and every | z Z Q \ ^ Q ; then for all these w's and #*s, | Pn (z)\ = | (1 +/ whl (*)) ... (1 +/ n (ar)) I ^ e I/ M+1 Wl + - + !/<*) I < 3. It follows precisely as on p. 382 that the series /Wl + (Pm+* Pm+l) + + (Pn ^n-l) + converges uniformly in | z Z Q \ ^ Q. As all the terms of this series are analytic in | z Z Q \ < r, the series itself, by 249, therefore represents a function F m (z) analytic in | # Z Q \ < g. Hence F(*) =- II (I +/ (*)) = (1 +A (*)) ... (1 +/, (*)) F m (z) w-1 is also an analytic function, regular in that circle. From the above considerations, we may deduce two further theorems, which provide an analogue to Weierstrass 1 theorem on double series: Theorem 1. With the assumptions of the preceding theorem, the ix-253. pansion in power series of F (z) may be obtained by expanding the product term by term. More precisely, we know that the (finite) product P* (*) = //(!+/,(*)) P=l may be expanded in a power series of centre Z Q which converges for | z ZQ | < r, since this is the case with each of the functions f lt / 2 , . . . . 15 (051) 438 Chapter XII. Series of complex terms. Let the expansion be P]c ( z -) = A > +A?\z-z ) + A! i ''\ Z -z }* + -. + A Then for each (fixed] n = 0, 1, 2, . . ., the limit lim A = X n &->+ exists, and *(*) = JT(i + 4 0) = 1 ^ (* - *)" A;=l n = Proof. By 46, theorem 2, the uniform convergence, in I z z I ^ , of the scries used in the preceding proof, implies the uniform convergence in the same circle of the series 48 PI W + [P. W - PX ()] + + [P k W - P*-, (*)] + Applying W^f^ys^rass' theorem on double series to this series, we obtain precisely the theorem stated. Finally we prove a theorem about the derived function of F(z], quite similar to 218,2: Theorem 2. For every z in \z - Z Q \ < r for which F(z)*^Q, we have *. . //t^ series on the right hand side converges for all these values of z and gives the ratio on the left hand side, the logarithmic dif- ferential coefficient of F(z). Proof. We saw that the expansion F(z) = PI(Z) + (P 9 (z) - P, (z)) + ... was uniformly convergent in \z z \^e<r- By 249, F'W = P/W + (P,'W-P 1 ' (*)) + -... which implies that PiW-^F'W at every point in the circle. If at a particular point F(z) =J- 0, we have P B (z) =^ for each n, and hence by 41, 11, 48 For the remainders of the latter scries only differ from those of the former in that they contain the common factor P m (*), which is a con- ttnuous function for every z in the circle \z * I SS i and hence is bounded in this closed circle. 57. Products with complex terms. 439 Since, however, *V(*) _ f ,!(*)_ * ~ ~il + M*)' this is precisely what our theorem asserts. Examples. 1. If 2 a* is any absolutely convergent series of constant terms, the product 254. 77(1 represents a function regular in the whole plane, by 252. By 253, its ex- pansion, in power series, which is convergent everywhere, is with Here the indices Aj , A 2 , . , ., A/ c independently take for their values all the natural numbers, subject only to the condition A t <C^a < <^&- The existence of the sums ^ 1} ^ 2 , . . . is secured by theorem 253 itself; it is also easy to verify that they are independent of the order of the terms. It was by applying this theorem that Euler * 9 and later C. G. J. Jacobi 60 were led to an abundance of most remarkable formulae. 2. We have where the product on the right hand side converges in the whole plane The proof is word for word the same as that given in 219, 1 for a real variable. 3. Taking z = i in the above sine product, we obtain XI t or (Cf. however the extremely easy evaluation of J[ \\ -J in 128, 6). 4. The sequence of functions n\n 2 converges for every z in the whole plane. In fact by 127, theorem 10, * 9 Introductio in analysin inf. Vol. 1, Chap. 15. 1748. 60 Fundamenta nova, Kbnigsberg 1829. 440 Chapter XII. Series ot complex terms. also, by 128, 2, the numbers y n = (1 -|- -- -f- H -- j log n tend, as n > -f OO , to Euler's constant C, so that the right hand expression, which is when divided by n z , tends to a definite limit as n * + OO. This proves the statement. Further, the limit y K(*) say, becomes only for z = Q, 1, - 2, .... Excluding these values, we have, for all other values of z, lim = lim H ~ This function of a complex variables (restricted only to be ={= 0, ~1 , 2, . . .) s the so-called G a WHHI- function F(z) which we have already defined on p. 385 for real values of the argument. We proceed to show that K(z) is analytic in the whole plane (i. e. an integral function). For this, it suffices to show that the series K to = ft to + fea W - ft <*)) H ----- h te (<*) - g n - 1 converges uniformly in every circle | z \ < Q . Now to - gn-l W = *l (l) [(l also a constant ^ exists B1 such that | g t , (z)\< A for every v -= 1, 2, 3, . . . and every | z \ <J g, and further, we may write (see p. 283 and p. 442, footnote 54) where |# n (*)| remains less than some constant B for every n = 2, 3, ... and 51 Let | z | < and n > w > 2 ^ . Then + . . . i + . : where log l+i = i + ^-. As <I (cf. p. 435) we have | r, v \ < |,| < ff and the last factor in the preceding expression therefore remains << e b = ,4 3 , for every |^|<^ and every w;> w. Similarly the last factor but one (see p. 295), also remains less than a fixed number A 9 . As the remaining factor is also always less than a fixed number A for every \z\ fj #> it follows that I * (?) I ^ ^i'^a'^8 f r a ^ these values of * and every n >> m . On the other hand, the first m functions | g 1 (z) \ , | g. 2 (z) \ , . . ., | ^ OT U) | also remain bounded for every | z \ < g> ; the existence of the number A as asserted in the text is thus established. If z is restricted to lie in a circle , in the interior and on the boundar}' ot which s=^0, 1, 2, ... and | z \ < Q , then for every n > m From this we infer in exactly the same way that a constant A' exists such that | ,-;- | < A' in ft, Cor every n = 1 , 2, 58. Special classes ot series of analytic functions. A. Dirichlet's series. 441 every | *[<(> Thus for all these z's and w's, /N.^ , I * 2 , #(-) . *'#nM I fi, ~ In- 1 W | < A . I - _ + -^L + - -V- where C is a suitable constant By 197, it follows that the series for K(z) converges uniformly in the circle |*|<e, indeed the series of absolute values 2 1 g n (z) g n -l (*) | does so, and, by 249, K(z) is analytic in the whole plane 58. Special classes of series of analytic functions. A. Dirichlet's series. A Dirichlet series is a scries of the form 52 Here the terms as exponential functions are analytic in the whole plane. The chief question will therefore be to determine whether and where the series converges and, in particular, whether and where it converges uniformly. We have Theorem 1. To every Dirichlet series there corresponds a real 255, number X known as the abscissa of conrerf/ence of the series such that the series converges when $R (z) > A and diverges when >H ( z) < A . The number I may also be oo or +00; in the former case the series converges everywhere, in the latter nowhere. Further, if jl =|= -|- oo and X>h, the series is uniformly convergent in every circle of the half -plane $ft (z) ^> /' and accordingly the series, by Weier- strass 1 theorem 249, represents a function analytic and regular in every sucn circle and hence in the half-plane 63 W (z) > A. The proof follows a line of argument similar to that used in the case of power series (cf. 93) We first show that if the series con- verges at a point Z Q> it converges at every other point z for winch 9t (z) > SR (* ). As however it suffices, by 184, 3 a, to show that the series A 1 n=i - 1 form 5<J iMore generally, a sories is called a Dirichlet series \\hen it is of the ~- m ~ or of the form ^a n e~^ nZ , where the p n 's are positive numbers and the A n 's any real numbers increasing monotonely to -f oo. 63 The existence of the half-plane of convergence was proved by /. L. W. V. Jensen (Tidskrift for Mathematik (5), Vol. 2, p. 63. 1884); the uniformity of the convergence and thereby the analytic character of the function represented were pointed out by E. Cohen (Annales fie. Norm. sup. (3), Vol. 11, p. 75. 1894; 412 Chapter XII. Series of complex terms, is convergent. Writing (for a fixed exponent (z Z ] ZD ,0, 4 i " n the numbers 0-n -> (* #<))> ^ ' s at once seen M J tne Y are therefore cer- tainly bounded, \ 8- B | < A, say. The th term of the above series is therefore and the series is accordingly convergent when ?R(z z )>0. As a corollary, we have the statement: If a Dirichlei series is divergent at a point z = z^ 9 it is divergent at every other point whose real part is less than that of z 1 . Supposing that a given Dirichlet series does not converge everywhere or nowhere, the existence of the limiting abscissa Jl is inferred (as in 93) as follows: Let z' be a point of divergence and z" a point of convergence of the series, and choose a? <SR(^) and y >5R(O> both real - For z = x o thc series will diverge, for z = y it will converge. Now apply the method of successive bisection, word for word as in 93, to the interval / = x . . . y on the real axis. The value i so obtained will be the required abscissa. Now suppose X > i (for i = oo, A' may therefore be any real number); if z is restricted to lie in a domain G in which JR^^Jl' and |2|f^-R, so that in general G will take the shape of a seg- ment of a circle, our series is uniformly convergent in that domain. To show this, let us choose a point Z Q for which A< 3l(z Q )<A'; as before, we write 54 More generally, we may at once observe that if |*| < - and \w\ < /?, Ct and if we write, taking the principal value, the factor &, which depends on z and w, remains less than a fixed constant for all the values allowed for z and w. Proof: (l-f-z) W -^ 10 ^^ = ^^^ a) , With iy= * +/---.fl+.... u o 4 For every | z \ < -^- , we therefore have 1 17 | < 1 ; hence in the expression in square brackets, which was denoted by #, satisfies the in- equality 1*1 <* 2jR - This is at once obvious if we replace all the quantities in thc brackets by theii 58. Special classes of series of analytic functions. A. Dirichlet's series. 443 V is a convergent series of constant terms; by 198, 3 a it there- n z fore suffices to show that 1 1 converges uniformly in the domain in question and that the factors - are uniformly bounded in G. Now, writing X 91 (z^) = d (> 0), 1 1 i*~* (n +!)*-* Using the evaluation given in the preceding footnote (or else directly, by expanding (1 -| J =0 in powers of (z - Z Q )) we now see that a constant A certainly exists such that the difference within the modulus signs on the right hand side of the above inequality is in absolute value <4 for every z in our domain and every n expression on the right is thus 1, 2, 3, The whole On the other hand, since T-, the factors are uniformly bounded in G. By 198, 3 a, this proves that the Dirichlet series is uniformly convergent in the domain stated, and hence, in particular, that every Dirichlet series represents a function which is analytic in the interior of the region of convergence of the senes (the half- plane 9tf (z) > X) . From 1 yr <*LJ it follows at once that if a Dirichlet series converges absolutely at a point Z Q , it does so at any point z for which 9^(2) >SK(2 ), and if it does not converge absolutely at Z Q) then it cannot do so at any point z for which 9fi (z) < SR (Z Q ) . Just as before we obtain Theorem 2. There exists a definite real number I (which may also be +00 or oo) such that the Dirichlet series converges ab- solutely for $l(z)>l, but not for ft(z)<l. Of course we have A<^Z; over and above this, the relative posi- tions of the two straight lines $R (z) = Jl and jR (2) = / is subject to the following 444 Chapter XII. Series of complex terms. Theorem 3. We have in every case I A<^1. Proof. If J is convergent and 9t (z) > 9ft (z ) + 1 , then a n is absolutely convergent, for with $1 (z ZQ) > 1 . This proves the statement at once. Remarks and Examples. . 1. If a Dirichlet series is not merely everywhere or nowhere convergent the situation will in general be as follows, the half-plane *)t (z) < A of divergence of the series is followed by a strip A < $ft (z) < I of conditional convergence of the series; the breadth of this strip is in any case at most 1, and in the remaining half-plane $R (#) > /, the series converges absolutely. 2. It may be shown by easy examples that the difference / A may assume any value between and 1 (both inclusive), and that the behaviour on the bounding lines 8t (z) = A and 9} (z) = / may vary in different cases. 1 2 n 3. The two series J?on~~ z anc * z provide simple examples of Dirichlet series which converge everywheie and nowhere. 4. 27 - has the abscissa of convergence A = 1 ; thus it represents an analytic n z function, regular in the half-plane fll (#) > 1. It is known as Riemann\ ^-function (v. 197, 2, 3) and is used in the analytical theory of numbers, on account of its connection with the distribution of prune numbers (see below, Rem. 9) 55 . 5. Just as the radius of a power series can be deduced directly from its co- efficients (theorem 94), so we may infer from the coefficients of a given Dirichlet series what positions the two limiting straight lines occupy. We have the following Theorem. The abscissa of convergence A of the Dirichlet series 2 n z is invariably given by the formula _ j l A = lim log a u+l -f a^ 2 -f- . . . + a v x where x increases continuously and [eW] =w, 0*] =;. Substituting a n for a n in this formula, we obtain I, the limiting abscissa of absolute convergence 66 . 0. A concise account of the most important results in the theory of Dinchlet's series may be found in G. H. Hardy and M. Riesz, Theory of Dirichlet's series, Cambridge 1915. 65 A detailed investigation of this remarkable function (as well as of arbitrary Dirichlet series) is given by E. Landau y Handbuch der Lehre von der Verteilung der Primzahlen, Leipzig 1909, 2 Vols., in E. Landau, Vorlesungen uber Zahlentheorie, Leipzig 1927, 3 Vols., and in E. C. Titchmarsh, The Zeta-Function of Riemann, Cam- bridge 1930. 56 As regards the proof, we must refer to a note by the author: "Uber die Abszisse der Grenzgeraden emer Dinchletschen Reihe" in the Sitzungsberichte der Berliner Mathematischen Gesellschaft (Vol. X, p. 2, 1910). 58. Special classes of series of analytic functions. A. Dirichlet's series. 445 7. By repeated term-by-term differentiation of a Dinchlet series F (*) ~ 2 ***> we obtain the Dirichlet series n - (fixed,). As an immediate consequence of Weierstrass' theorem on double scries, these neces- sarily cannot have a larger abscissa of convergence than the original series, and, owing to the additiopal factors log" n, they can obviously not have a smaller one either. They represent, in the interior of the half-plane of convergence, the derived functions F< v) (z). 8. By 255, the function represented by a Dinchlet series can be expanded in a power series about any point interior to the half-plane of convergence as centre. The expansion itself is provided by Weierstrass' theorem on double co 1 series. If, for instance, it is required to expand the function (*) = *=1 k z about Z Q = -j~ 2 as centre, we have for k = 2, 3, ... and this continues to hold for k = 1 provided we interpret (log lj as having the value 1. Hence for n>0 - which gives the desucd expansion 9. 1'or (*)>!, CO J J the series VJ and the product 77 - n=in 2 ' JI l-/>' 2 (where p takes for its values all the prime numbers 2, 3, 5, 7, . . in succession) have everywhere the same value, and accordingly both represent the Riemann - func- tion (z). (Euler, 1737; v. Introd. in analysin, p. 225) Proof. Let z be a definite point such that 5H (z) = 1 -f-<5 > 1 . By our remark 4 and 127, 7, the series and product certainly converge absolutely at this point. We have only to prove that they have the same value. Now multiplying these expansions together, for all prime numbers p< AT, _ whore AT denotes an integer kept fixed for the moment, the (finite) product so obtained is where the accent on the 2"* indicates that only some, and not all, of the terms of the series written down are taken. Here we have made use of the elemen- tary proposition that every natural number > 2 can be expressed in one and only one way as a product of powers of distinct primes (provided only positive 15 * (G5l) 446 Chapter XII. Series ot complex terms. integral exponents are allowed and the order of succession of the factors is left out of account). Accordingly 7T * V> V On the right hand side \ve have the remainder of a convergent series, which tends to when N - + co. This proves the equality of the values of the infinite product and of the infinite series, as was required. 10. By 257, we have for ffi (z) > 1 where H (1) = 1, M (2) = - 1, p (3) = - 1, p (4) = 0, p (5) - - 1, p (6) = + 1, . . . and generally /x (w) 0, + 1 , or 1 according as n is divisible by the square of a prime number, or is a product of an even number of primes, all different, or of an odd number of primes, all different. The product-expansion of the ^-function also shows that for JR (#) > 1, we always have (z) =t= 0. The curious coefficients fi (n) are known as Mobius* coefficients. There is no superficial regularity in the mode of succession of the values 0, -f- 1, 1 among the numbers /LI (n). 11. Since f (z) - ^ converges absolutely for 81 (?)>!, we may form n z the square (f(*)) 2 by multiplying the series by itself term by term and re- arranging in order of increasing denominators (as is allowed by 91). We thus obtain where i n denotes the number of divisors of n. These examples may suffice to explain the importance of the - function in problems in the theory of numbers. B. Faculty series. A faculty series (of the first kind) is a scries of the form / n V ( ' which of course has a meaning only if 2 + 0, 1, 2, .... The questions of convergence, elucidated in the first instance by Jensen, are completely solved by the following 258. Theorem of />/.rfu 67 . The faculty series (F) converges with the exclusion of the points 0, 1, 2, ... wherever the "asso- ciated" Dirichlet series ^ converges, and conversely the latter converges wherever the series (F) con- verges. The convergence is uniform in a circle for either series , when it is so for the other, 'provided the circle contains none of the points 0, 1, 2, ... either in its interior or on its boundary. 6 J Uber die Grundlagen der Theorie der Fakultatenreihen. MUnch. Ber Vol. 36, pp. 151218. 1906. 58. Special classes of series of analytic functions. B. Faculty series. 447 Proof. 1. We first show that the convergence of the Dirichlet series at any particular point =j= 0, 1, 2,... involves that of the faculty series at the same point. As >...(*+ n) n * g n (z)' if g n (z) has the same significance as in 1354, example 4, it is sufficient, by 184, 3 a, to show that the series n=l &(*) 8* + i(*) n =l I &W'& + i (*) I is convergent Now tends to a finite limit as w increases, namely Sn (2) t> the value F(z); hence, in particular, this factor remains bounded for all values of w (z being fixed). Hence it suffices to establish the con* verge nee of the series But this has been done already in 254, example 4. 2. The fact that the convergence of the faculty series at any point invoK es that of the Dirichlet series follows in precisely the same manner, as again, by 184, 3 a, everything turns on the convergence of -!&,(*) -ft,+il- 3. Now let be a circle in which the Dirichlet series converges absolutely and which contains none of the points 0, 1, 2, . . ., either as interior or boundary points. We have to show that the faculty scries also converges uniformly in that circle. By 198, 3 a, this again reduces to proving that a fg\.a (z Bn \ z ) * Sn + 1 \*. is uniformly convergent in $ and that the functions 1 / g n (z) remain uniformly bounded in . The uniform convergence of n=l was already established in 254, 4. Also it was shown on p. 440, footnote 51, that there exists a constant A' such that 1 for every z in Jf and every w. This is all that is required. (Cf. -H>, theorem 3.) 4. The converse, that the Dirichlet series converges uniformly in every circle in which the faculty scries does so, follows at once by 198, 3 a from the uniform convergence of the series 2 \ g n + ,() g n (2) | and the uniform boundedness of the functions g n (z) in the circle, both of which were established in 254, 4. 448 Chapter XII. Series of complex terms. Examples. 1. The faculty series S 1 ' converges at every point of the plane =(=0, 1, ... . For the Dirichlet series 00 I z 1 n=l is evidently convergent everywhere. As .111 1 x x x+l x(x+l)' /I* 1 - 2! A* 1 kl A T /.. 7~iT~/ . " ?i\ > *> /3 - ' = _ _ _ _ _ x *(*+l)(ar + 2) f ""' x x (x -f 1) - - (x + *) ' the given faculty series results simply, by Euler's transformation 144, from the series ~~ To show this, we have only to subtract the terms of the right hand side sue cessively from the left hand side. After the w th subtraction we have n\ 1 n\n z 2. It is also easily seen (cf. pp. 2656) that for $R (z) > 1 01 . _ 11 __ , . (*-!)! """"" ). ..(*+) z-n* * and this, by 254, example 4, tends to when w-*oo, provided (Stirling Methodus differentialis, London 1730, p. 6 seqq.) C. Lambert's series. A Lambert series is a series of the form 58 If we again inquire what is the precise region of convergence of the series, it must first be noted that for every z for which z n 1 can be equal to zero, an infinite number of the terms of the series be- come meaningless. For this reason, the circumference of the unit circle will be entirely excluded from consideration 59 while we discuss the 68 A more extensive treatment of this type of series is to be found in a paper by the author: Cber Lambertsche Reihen. Journ. f. d. reine u. angew. Mathem., Vol. 142, pp. 283315. 1913. 59 This does not imply that this series may not converge at some points ar t of this circumference, for which zf =t= -f 1 for every n *Z 1. This may actually happen; but we will not consider the case here. 58 Special classes of series of analytic functions. C. Lambert's series. 449 question of convergence of these series, and the points inside and outside the circle will be examined separately. We have the following theorem, which completely solves the question of convergence in this respect: Theorem. // 2a n converges, the Lambert series converges for every z 259. whose modulus is 4 s ! W ^ a n * s n t convergent, the Lambert series converges at precisely the same points as the "associated" power series a n z n provided \ z \ + 1 as before. Further, the convergence is uniform in every circle & which lies completely (circumference included} within one of the regions of convergence of the series and contains no point of modulus 1. Proof. 1. Suppose 2a n divergent. The radius r of 2 a n z n is in that case necessarily <^ 1 and we have to show first that the Lambert series and the assoaated power series converge and diverge together for every \z\ < 1, and that the Lambert series diverges for \z\ > 1. Now y w = y z " .(\ n \ and vi ** v-. n 1 Accordingly, it suffices, by 184, 3 a, to establish the convergence of the two series and I 7* I - M _ L V I* I for |s|<l. The first of these facts is obvious, however, while the second follows from the remark that for | z \ < 1, we have 1 1 z n \ > -x- for all sufficiently large w's. On the other hand, if the Lambert series converged at a point z v where | Z Q | > 1 , the power series would converge for z = 2 , and by 93, theorem 1, would have also to converge for z = + 1. Hence the series would also have to converge, which is contrary to hypothesis. Finally, the fact that the Lambert series converges uniformly in \z\ <* Q <. r may at once be inferred from the corresponding fact in the case of the power series \a n z n \, by 46,2, in virtue of the inequality 450 Chapter XII. Series of complex terms. The case where 2a n diverges is thus completely dealt with. 2. Now suppose a n convergent, so that 2 a n z n has a radius ?^> 1. The Lambert series is certainly convergent for every | z \ -< 1 and indeed uniformly so for all values of z such that | z \ <^ > < ! For I z I ^> Q f > 1 , we have and as I .-6T-, "!_(!)' z ' < 1, this reduces the later assertions to the pre- ceding ones, and the theorem is therefore established in all its parts. By the above, a very simple connection exists, in the case where _T a n is com ergent, between the sum of the series at a point z outside the unit circle and the same sum at the point inside it. Accordingly it will suffice if we consider only thdt region of convergence of the series which lies inside the unit circle. This is either the circle | z \ < ? or the unit circle | z \ < 1 itself, according as the radius r of the series 2 a n z n is < 1 or ^> 1. Let r 1 denote the radius of this perfectly definite region of convergence. The terms of a Lambert series are analytic functions regular in l^l^fj, and for e\ery positive Q <: r lf the series is uniformly con- vergent in | z \ <LQ'- hence we may apply Weierstrass* theorem on double series to obtain the expansion in power series of the function re- presented by a Lambert series m | z \ < r . We have + a a ** + a,2 + and we may add all these series together term by term. In the th row, a given power z n will occur if, and only if, n is a multiple of k, or k a divisor of n. Therefore A n , the coefficient of z n in the result- ing series, will be equal to the sum of those coefficients a v whose suffix r is a divisor of n (including 1 or w). This we write sym- bolically 60 A n = a d , d/n and we then have, for | z \ < r l , In words: the sum of all a d 's for which d is a divisor of n. 58. Special classes of series of analytic functions C. Lambert's series. 451 Examples. 26O. 1. a n = 1. Here A n is equal to the number of divisors of n t which (as in 257, example 11) we denote by r n \ then = z -}- 2 z* + 2 as 3 -f 3 s 4 + 2 # 5 + 4 z* -f 2 3 7 -f 4 z* -f . . . . In this curious power series, the terms # n whose exponents are prime numbers are distinguished by the coefficient 2. It was due to the misleadmgly close connection between this special Lambert series and the problem of primes that this series (as a rule called simply the Lambert series) 61 played a considerable part in the earlier attempts to deal with this problem. But nothing of importance was obtained in this manner for some time. Only quite recently AT. Wiener 62 succeeded by this means in proving the famous prime number theorem. 2. a n = n. Here A n is equal to the sum of all the divisors of n, which we will denote by r n '. Thus for | z \ < 1 E n T *---.= J7T n 's = sr+3s 8 + 4* 8 + 7* + 6* + 12*' + ---- n 1 L ~~ z n=\ 3. The relation A n Za d 1S uniquely reversible, i. e. for given A n 's, the din coefficients a n can be determined in one and only one way so as to satisfy the relation. We then have in fact where /i (k) denotes the Mobtui coefficients defined in 257, example 10, whose values are 0, -|- 1 and 1. In consequence of this fact, not only can a Lambert series always be expanded in a power series, but conversely every power series may be expressed as a Lambert series, provided it vanishes for z = 0, i. e. A - 0. But it should be observed that a relation of the form need not remain true for | 3 \ > 1, even when both series converge there. 4. For instance, if A = 1 and every other A n = 0, a n - /* (), and we have the curious identity =. J7 /iW r S \n <I 71=1 * ~~ ~ 6. Similarly, we find the representation, valid for | 2 | < 1, where 9 (n) denotes the number of integers less than n and prime to , a number introduced by Euler. oo ^ n on 6. Writing Z a n .- ^~ n = f(z) and Z a n z n g (z), and grouping the terms n 1 X ~ n 1 by diagonals in the double expansion of the Lambert series on p. 450 (which is allowed), we obtain /(*)*(*) + *<*) + ...- 27 *("). 61 Lambert, JH. Anlage zur Architektonik, Vol. 2, p. 507. Riga 1771. 62 Wiener, N., a new method in Taubenan theorems, J. Math. Massachusetts, Vol. 7, pp. H>1184, 1928, and Taubenan theorems, Ann. of Math. (2), Vol. 33, pp. 1100, 1932. 452 Chapter XIL Series of complex terms. 1 (_i)n-i 7. bor a n = ( I)* 1 " 1 , = n, = ( I)"" 1 , =--,== v ^ , = a n , . . . , we n \ / , v / ' w n obtain in this way, successively, the following remarkable identities, valid for | # | < 1, in which the summations are taken from n 1 to oo : z n z n a) 2 ( l) n ~ L .-__ ^ --- 2 1 7 n > b > ^"i^h ^ -^(1 -^i* e) 27 - ' " --- Z*log(l 4- ^ n ), W 1 l) ^ Qt 1 ~ === /^/ i n (I OC I <C 1), etc. 8. In the two identities d) and e) we have on the right hand side a series of logarithms (for which of course we take the principal values) ; thus simple con- nections can be established between certain Lambert series and infinite products. E. g. from the two identities in question: ^.1 z n y II (1 h *") = ", with - Z 1 - -' , . 71 I / 9. As an interesting numerical example we may mention the following: Taking M O -- 0, HI I, and for every n > 1, */ n w n _i + w n -2 w ^ obtain Fibonacci's sequence (cf. 6, 7) 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, .... We then have Z l i. 1 .!*... 1 . 1 . A= f r /3 - V'5\ r /7 - 3 A ,-* * + 3 + 8 + 21 + 55 + ' - - V5 [L (- -V J - , ^ ._ __ JC n where L (je) denotes the sum of the Lambert series 3 , _ n . The proof is based on the fact, which is easily established, that x where a and ft are the roots of the quadratic equation x 2 x 1 = 0. (Cf. Ex. 114.) Exercises on Chapter XII 64 . 174. Suppose z n -> and b n -> b 4= 0. Under what conditions may W3 infer that bj* -* tfl 175. Suppose z n -> X) (i. e. | z n | -> -f- oo). Under what conditions may we then infer that a) (l + -)* -> e*, \ z n / _ b) z n - (z l l s n - 1) -> log *? es Landau, E.: Bull, de la Soc. math, de France, Vol. 27, p. 298. 1899. 64 In these exercises, wherever the contrary does not follow clearly from the context, all numbers are to be regarded as complex. Exercises on Chapter XII. 453 176. The principal value of z* remains, for all values of z t less in absolute value than some fixed bound. 177. If z n = j?(- If (*), v \ v / either z n -> or -> 0, according as 9R (z) > or < 0. What is the behaviour s n of Csy n ) when 9R (z) -= 0? 178. Let 0, A, c, </ be four constants for which a d b c ^ and let # be arbitrary. Investigate the sequence of numbers (z , z lt z 2t ) given by the re- currence formula z LI a ~ n ~- (n 1 2 > n hi _ _ i V v J > ^ / C5r n -t- a What are the necessary and sufficient conditions that (z n ) or ( ) should converge? \xr n / And if neither of the two converges, under what conditions can z p become z again for some index />? When are all the -3r n 's identically equal? 179. Let a be given 4= and Z Q chosen arbitrarily, and write for each n ^ 1 (z n ) converges if, and only if, Z Q does not lie on the perpendicular to the straight line joining the two values of V a through its middle point. If this condition is fulfilled, (z n ) converges to the value of V a nearest to sr . What is the behaviour of (x n ) when s lies on the perpendicular in question? 180. The scries 27- {+ ~ does not converge for anv real y ; the series 27 TV---- , -- , ~* n ~ " log n on the other hand, does converge for every real y =4= 0. 180 a. The refinement of Weierstr ass's theorem 228 that was mentioned in footnote 13, p. 399, may be proved as follows in connection with the foregoing example: From the assumptions, it follows, firstly, that we may write (n ^^r - - l + & (y - Min (A> 2) > X) ' where the B n 's are bounded; hence, secondly, that we may write (C N"" 1 1 -H -~x ~ ) satisfy the as- sumptions of the test 184, 3. If 2 a n were to converge, then 2 a n b n = E - would also have to converge, contrary to the preceding example and theorem 255. 181. For a fixed value of z and a suitable determination of the logarithm, does tend to a limit as n -> -|-QO? 182. For every fixed x with < SR (z) < 1, lim 1 + tt , -h -, + ...+ exists (cf. Ex. 135). 183. The function (1 z) - sin Hog = 1 may be expanded in a power series 27 a n z n for | z | < 1 , if we take the principal value for the logarithm. Show Jhat this senes stjjl converges absolutely for J z J 1. 454 Chapter XII. Series ot complex terms. 184. If x tends to + 1 from within the unit circle, and "within the angle", we have a) l_, + .,_,,<> + ,t_4. _!.. b) (l- . ..- _ d) (1-, . 2 a n z n ,. a n e) y * --*lim 5 ^&* n b n provided the right hand limit exists, b n is positive for each w, and Sb n is divergent. 185. Investigate the behaviour of the following power series on the circumference of the unit circle: V 1 ( ^) n v ** n ' tt + f e ) 2i f" 2 *"> wnere n has the same meaning as in Ex. 47. 186* If S a n z n converges for | z \ -< 1 and its sum is numerically for all such values of *, then ^|a n | a converges and its sum is < 1 . 187. The power series h) J-__ all have the unit circle as circle of convergence. On the circumference, they also converge in general, i. e. with the possible exception of isoLited points. Try to express their sums by means of closed expressions involving elementary functions; separate the real and imaginary parts by writing 2 = 7 (cos x -\-i sin x), and write down the trigonometrical expansions so obtained for r < 1 and for r = 1 separately. For which values of x do they converge? What are their sums? Are they the Fourier series of their sums? 188. What are the sums of the following series: Y7 cos nx cos n y coswccsinwy H ft . -_. sin n x sin ny f\ X' ^, / -*-/ M rl and of the three further series obtained by giving the terms of the above series the sign (- 1) B ? tixercisee on Chapter XII. 455 189. Proceeding with the geometric series z* as in Ex. 187, but leaving 1 , we obtain the expressions 1 r cos x a) V r n cos nx = - b) >7 ?" sin n a; -, - --- ; 1 2 r cos x-}-r* rsma; = - 1 Deduce from them the further expansions f costta; w ^i (2 cos x) n and indicate the exact intervals of validity. 19O. In Exercise 187 a the following- expansion will have been obtained, among others. r n . f rsin x \ y. sin nx tan ~ x I . ^Tj n \ 1 r cos x J Deduce from it the expansions 00 ( & \ y, ( 1)" "* r n sm n x- sinn a; = tan - x (r -f- cot x) f -5- x\ , n=L and determine the exact intervals of validity. 191. Determine the exact regions of convergence of the following series [log log n] where (#>) is real and increases monotonely to +OO. 1!>2. Establish the relations *" -' l +* where the summation begins with n = l. 45G Chapter XII. Series of complex terms. 193. Corresponding to Landau's theorem (258) we have the following: The Dirichlet series 27 (~ I)"" 1 " and the so-called binomial coefficient series 2a n ( are convergent and divergent together, the points z = 1, 2, 3, . . . being disregarded. 194. For which values of z does the equation hold good? 195. Determine the exact regions of convergence of the following in finite products: 77(1 196. Determine, by means of the sine product, the values of the products a) //(l+J*), *>//(!+) 0)^(1 + 5). for real values of a;. The second of these has the value 2~ 2 g [cosh (jtx^ 2) cos (jr x \/ 2) ] . Does this continue to hold for complex values of xl 197. The values of the products 195, i) and k), can be determined in ihe form of a closed expression by means of the F- function. 198. For 1 1 1< 1 , 199. By means of the sine product and the expansion of the cotangent in partial fractions, the following series and product may be evaluated in the + < form of closed expressions; x and y are real, and the symbol 2 f( n ) indicates n= oo -f CO +00 the sum of the two series 2 f( n ) and 2 f(-~ k )> & nd similarly for the n=0 k=l product: + 00 J +00 a) *' b) - <0 2 n=s- g 59. General remarks on divergent sequences. 457 Chapter XIII. Divergent series. 59. General remarks on divergent sequences and the processes of limitation. The conception of the nature of infinite sequences which we have set forth in all the preceding pages, and especially in 8 11, is of compai ativdy recent date; for a strict and irreproachable construction of the theory could not be attempted until the concept of the real number had been made clear. But even if this concept and any one general convergence test for sequences of numbers, say our second main criterion, were recognized without proof as practically axiomatic, it nevertheless remains true that the theory of the convergence of infinite sequences, and of infinite series in particular, is far more recent than the extensive use of these sequences and series, and the discovery of the most elegant results of the subject, e. g. by Euler and his con- temporaries, or even earlier, by Leibniz, Newton and their contem poraries. To these mathematician?, infinite series appeared in a very natural way as the result of calculation, and forced themselves into notice, so to speak : e. g. the geometric series 1 -\- x -\- x* -\- oc- curred as the non-terminating result of the division l/(l x); Taylor's series, and with it almost all the series of Chapter VI, resulted from the principle of equating coefficients or from geometrical considerations. It was in a similar manner that infinite products, continued fractions and all other approximation processes occurred. In our exposition, the symbol for infinite sequences was created and then worked with; it was not so originally, these sequences were there, and the question was, what could be done with them. On this account, problems of convergence in the modern sense were at first remote from the minds of these mathematicians 1 . Thus it is not to be wondered at that Euler, for instance, uses the geometric series i-y-^x-f-*/ -, \ X even lor x = 1 or x = 2, so that he unhesitatingly writes 2 1 Cf. the remarks at the beginning of 41. a This relation is used by James Bernoulli (Posit, arithm., Part 8, Basle 1696) and is referred to by him as a "paradoxon non inelegans". For details of the violent dispute which arose in this connection, see the work of R. Reiff men- tioned in 69,8. 458 Chapter XIII. Divergent series. or 1 - 2 + 2 2 - 2 J + ... = ^ ; similarly from f^_ -) = 1 + 2 x + 3 x 2 + . . . he deduces the relation i-a + 3-*+-... = i; and a great deal more. It is true that most mathematicians of those times held themselves aloof from such results in instinctive mistrust, and recog- nized only those which are true in the present-day sense 3 . But they had no clear insight into the reasons why one type of result should be admitted, and not the other. Here we have no space to enter into the very instructive discussions on this point among the mathematicians of the 17 th and 18 th centuries 4 . We must be content with stating, e. g. as regards infinite series, that Euler always let these stand when they occurred naturally by expanding an analytical expression which itself possessed a definite value 5 . This value was then in every case regarded as the sum of the series. It is clear that this convention has no precise basis. Even though, for instance, the series 1 1 + 1 1 H ... results in a very simple manner from the division 1/(1 x) for x 1 (see above), and there- fore should be equated to ^ there is no reason why the same series should not result from quite different analytical expressions and why, in view of these other methods of deducing it, it should not be given a different value. The above series may actually be obtained, for x = 0, from the function f(x) represented for every x > by the Dirichlet series f(*\ = J? ir-J)-" 1 = i_L4_!_!-L J{) -l "* 2*t-3- 4* -*-' or from t + putting x = 1. In view of this latter method of deduction, we should have 2 to take 1 1 + . . . = g, and in the case of the former there is no im- mediate evidence what value /(O) may have; it need not at any rate be + - . 2 3 Thus d'Alembert says (Opusc. Mathem., Vol. 5, 1768, 35; M^moire, p. 183): "Pour moi, j'avoue que tous les raisonnements et les calculs fond^s sur des series qui ne sont pas convergentes ou qu'on peut supposer ne pas 1'fitrc, me paraitront toujours tr&s suspects". 4 For details, see R. Reiff, loc. cit. 5 In a letter to Goldbach (7. VIII. 1745) he definitely says: ". . . so habe ich diese neue Definition der Summe einer jeglichen seriei gegeben: Summa cujusque seriei est valor expressions illius finitae, ex cujus evolutione ilia series oritur". 59. General remarks on divergent sequences. 459 Eulers principle is therefore insecure in any case, and it was only Enter* unusual instinct for what is mathematically correct which in general saved him from false conclusions in spite of the copious use \vhich he made of divergent series of ihis type 6 . Cauchy and Abel were the first to make the concept of convergence clear, and to renounce the use of any non-convergent series; Cauchy in his Analyse algebrique (1821), and Abel in his paper on the binomial series (1826), which is expressly based on Cauchy s treatise. At first both hesitated to take this decisive step 7 , but finally resolved to do so, as it seemed unavoidable if their reasoning were to be made strict and free from gaps. We are now in a position to survey the problem from above, as it were; and the matter at once becomes clear when we remember that the symbol for an infinite sequence of numbers in whatever form it is given, sequence, series, product or otherwise has, and can have, no meaning whatever in itself) but that a meaning was only assigned to it by us, by an arbitrary convention. This convention consisted firstly in allowing only convergent sequences, i. e. sequences whose terms approached a definite and unique number in an absolutely de- finite sense; secondly, it consisted in associating this number with the infinite sequence, as its value, or in regarding the sequence as no more than another symbol (cf 41, 1) for the number. However ob- vious and natural this definition may be, and however closely it may be connected with the way in which sequences occur (e. g. as suc- cessive approximations to a result which cannot be obtained directly), a definition of this kind must ne\ ertheless in all circumstances be con- sidered as an arbitrary one, and it might even be replaced by quite different definitions. Suitability and success are the only factors which can determine whether one or the other definition is to be preferred; in the nature of the thing itself, that is to say, in the symbol (s w ) of an infinite sequence 8 , there is nothing which necessitates any preference. We are therefore quite justified in asking whether the compli- cation which our theory exhibits (in parts at least) may not be due 6 Cf. on the other hand p. 133, footnote 6. 7 So far as Cauchy is concerned, cf. the preface to his Analyse algibrique, in which, among other things, he says: "Je me suis vu force d'admettre plusieurs propositions qui paraitront peut-ctre un peu dures, par exemple qu'une serie diver- gente n'a pas de somme". As regards Abel, cf. his letter to Holmboe (16. I. 1826) f in which he says: "Les series divergentes sont, en general, quelque chose de bien fatal, et c'est une honte qu'on ose y fonder aucune demonstration". As already mentioned (p. 458, footnote 3), J. d'Alembert had expressed himself in a similar sense as early as 1768. 8 (s n ) may be assumed to be any given sequence of numbers, in particular, therefore, the partial sums of an infinite series 27 a n or the partial products of an infinite product. We use the letter s, with its reminder of the word "sum", because infinite series are by far the most important means of defining sequences. 460 Chapter XIIT. Divergent series. to our interpretation of the symbol (s n ), as the limit of the sequence, assumed convergent, being an unfavourable one, however obvious and ready- to-hand it may appear. Other conventions might be drawn up in all sorts of ways, among which more suitable ones might per- haps be found. From this point of view, the general problem which presents itself is as follows: A particular sequence (s w ) is defined in some way, either by direct indication of the terms, or by a series or product, or otherwise. Is it possible to associate a "value" s with it, in a reasonable way? "In a reasonable way" might perhaps be taken to mean that the number s is obtained by a process closely connected with the previous concept of convergence, that is to say, with the formation of lims w s. This has been found so extraordinarily efficacious in all the preceding that we will not depart from it to any considerable extent without good reasons. "In a reasonable way" might also, on the other hand, be inter- preted as meaning that the sequence (s n ) is to have such a value s associated with it that wherever this sequence may occur as the final result of a calculation, this final result shall always, or at least usually, be put equal to s. Let us first illustrate these general statements by an example. The series 5862. 2:(-l)" = l-l+l-l+-..., i. e. the geometric series 2x n for x = 1, or the sequence (sjel, 0, 1, 0, 1, 0, ..., has so far been rejected as divergent, because its terms s n do not approach a single definite number. On the contrary, they oscillate unceasingly between 1 and 0. This very fact, however, suggests the idea of forming the arithmetic means c 9 S n Since S|| = ~ [1 + ( l) n ], we find that i + C- p.] ' ~"~ 2 (n + 1) 2 ' 4 (n + 1) ' so that s n f (in the former sense) approaches the value -^ : By this very obvious process of taking the arithmetic mean, we have accordingly managed, in a perfectly accurate way, to give a meaning to Euler's paradoxical equation 11 + 1 -- [-... = --., to 59. General remarks on divergent sequences. 461 associate with the series on the left hand side the number ^ as its LI "value", or to obtain this number from the series. Whether we can always equate the final result of a calculation to whenever it ap- Ct pears in the foim (- l) n , cannot of course be determined off-hand. In the case of the expansion ^ --- = 2x n for x = 1, it is certainly i ~~ x / _ j\n 1 so; in the case of - - ~ for x = 0, it is equally true, as may be shown by fairly bimple means (cf. Exercise 200); and a great deal more evidence can be adduced to show that the association of the sequence 1, 0, 1,0, 1, ... with the value obtained in the manner Lt described above is "reasonable" 9 . We might therefore, as an experiment, make the following de- finition. If, and only if, the numbers c ' __ S Q + *i -f ... + * , __ , 9 . s n -- ---- ~ ~~ ' " tend to a limit s in the previous sense, the sequence (S M ), or series 2 a n > will be said to "converge" to the "limit", or "sum", s. The suitability of this new definition has already been demon- strated in connection with the series 2( l) n , which now becomes convergent "in the new sense", with the sum -=-, which seems j thoroughly reasonable. Two further remarks will illustrate the ad- vantages of this new definition: 1. Every sequence (s n ), convergent in the former sense and of limit s, is so constituted, in virtue of Cauchy's theorem 43, 2, that it would also have to be called convergent "in the new sense", with the same limit s. The new definition would therefore enable us to accomplish at least all that we could do with the former, while the example of the series -Z*( l) n shows that the new definition is more far-reaching than the old one. 2. If two series, convergent in the old sense, -2*0 n = A and 2b n = B, are multiplied together by Cauchys rule, giving the series 2c n E 2(a Q b n + !&-.! H ----- t-0 n fy))> we k now tli at mis series is not necessarily convergent (in the old sense). And the question when c n does converge presents very considerable difficulties and has not been satisfactorily cleared up so far. The second proof of theorem 189, 9 From the series (see above) for - - - ^ also we can accordingly de- 2 duce the value ~ for x = 1 . We have only to observe that the series, written o somewhat more carefully, is 1 4- 0-a; x- -\- X 9 + 0# l a; 5 -f--l -- , and is therefore l + O l + l + O l+H ----- for a; = J f 4:62 Chapter XIII. Divergent series, however, shows that in every case if C n denotes the n th partial sum of 2c n . The meaning of this is that <Sc n always converges in the new sense, with the sum AB. Here the advantage of the new convention is obvious: A situation which, owing to the insuperable difficulties involved, it was impossible to clear up as long as we kept to the old concept of convergence, may be dealt with exhaustively in a very simple way, by introducing a slightly more general concept of convergence. We shall very soon become acquainted with other investigations of this kind (see 61 in particular); first of all, however, we shall make some definitions relating to several fundamental matters: Besides the formation of the arithmetic mean, we shall become acquainted with quite a number of other processes, which may with success be substituted for the former concept of convergence, for the purpose of associating a number s with a sequence of numbers (s n ). These processes have to be distinguished from one another by suitable designations. In so doing it is advisable to proceed as follows: The former concept of convergence was so natural, and has stood the test so well, that it ought to have a special name reserved for it. Accor- dingly, the expression: "convergence of an infinite sequence (scries, product, . . .)" shall continue to mean exactly what it did before. If by means of new rules, as, for instance, by the formation of the arithmetic mean described above, a number s is associated with a sequence (s n ), we shall say that the sequence (s n ) is limitable* by that process, and that the corresponding series 2a^ is summable by the process, and we shall call s the value of either (or in the case of the series, its sum also). When, however, as will occur directly, we are making use of several processes of this kind, we distinguish these by attached initials A, B, . . ., F, . . ., and speak for instance of a F- process 10 . We shall say that the sequence (s n ) is limitable F, and that the series 2a n is summ able F; and the number 5 will be referred to as the Flimit of the sequence or Fsum of the series; symbolically F-lim s n == s, V-2a n = s . When there is no fear of misunderstanding, we may also express the * German: hmitierbav. 10 In the case of the concept of integrability the situation is somewhat similar and it was prob.ibly in this connection that the above type of notation was first introduced. 'Ihus we say a function is integrable /? or integrable L according as we are referring* to integrability in Riemann's or in Lebesgtte's sense. 50. General remarks on divergent sequences, 463 former of the two statements by the symbolism which more precisely implies that the new sequence deduced from (s n ) by the F-process converges to s. When, as will usually be the case in what follows, the process admits of a &-fold iteration, or can be graded into different orders, we attach a suffix and speak of a V ^-limitation process, a V ^-summation process, etc. In the construction and choice of such processes we shall of course 263. not proceed quite arbitrarily, but we shall rather let ourselves be guided by questions of suitability. We must give the first place to the fundamental stipulation to be made in this connection, namely that the new definition must not contradict the old one. We accordingly stipulate that any F- proccss which may be introduced must satisfy the following permanence condition : I. Every sequence (s n ) convergent in the former sense, with the limit s, must be limitable V with the value s. Or in other words, lim s n = s must in every case imply u F-lim s n s. In order that the introduction of a process of this kind may not be superfluous, we further stipulate that the following extension con- dition is to hold: II. At least one sequence (sj, which diverges in the former sense, must be limitable by the new process. Let us call the totality of sequences which are limitable by a particular process the range of action of this process. The condition II implies that only those processes will be allowed which possess a wider range of action than the ordinary process of convergence. It is precisely the limitation of formerly divergent sequences and the summation of formerly divergent series which will naturally claim the greater part of our attention now. Finally, if several processes are employed together, say a V- process and a HP- process simultaneously, we should be in danger of hopeless confusion if we did not also stipulate that the following compatibility condition should be fulfilled: III. // one and the same sequence (s n ) is limitable by two different processes, simultaneously applied, then it must have the same value by both processes. In other words, we must in every case have = W'\ims n , if both these values exist. 11 We might also be satisfied if some convergent sequences at least are limitable with unaltered value by the process considered. This is the case e. g with the E f - process discussed further on, provided the sutfix p is complex 464 Chapter X11I. Divergent series. We shall only consider processes which satisfy these three con ditions. Besides these, however, we require some indication whether the association of a value 5 with the sequence (sj effected by a parti- cular V- process is a reasonable one in the sense explained above (p. 460). Here widely -varying conditions may be laid down, and the processes which are in current use are of very varied degrees of ef- ficiency in this respect. In the first instance we should no doubt require that the elementary rules of the algebra of convergent sequences (v. 8) should as far as possible be maintained, i. e. the rules for term-by-term addition and subtraction of two sequences, term-by-term addition of a constant, and term-by-term multiplication by a constant, and the effect of a finite number of alterations (27, 4), etc. Next we might perhaps require that if, say, a divergent series 2a n has associated with it the value s, and if this series is deduced, e. g. from a power series f(x) = 2c n x n by substituting a special value X L for x, then the number 5 should bear an appropriate relation to f(x^) or to Mmf(x) for x^x^\ and similarly for other types of series (Dirichlet series, Fourier series etc.). In short, we should require that wherever this series appears as the final result of a calculation, the result should be s. The greater the number of conditions similar to the above which are satisfied by a 264. particular process let us call them the conditions F, without taking pains to formulate them with absolute precision and at the same time, the greater the range of action of the process, the greater will be its usefulness and value from our point of view. We proceed to indicate a few of these processes of limitation which have proved their worth in some way or another. 205. v 1. The C r , H^ 9 or Af-process 12 . As described above, 262, we form the arithmetic means of the terms of a sequence (s n ): 0.1.2,...) which we will denote by c n \ h n ', or m n . If these tend to a limit s in the older sense, when n -> oo, we say that (s n ) is limit able C or limitable H l or limit able Af with the value s and we write Af-lim s n = s or M (s n ) or use the letters C x or H^ instead of M. The series Z a n with the partial sums s n will be called summable C l or summable H l or summable Af, and s will be called its C^-, //!-, or M-sum. The sequence of units 1, 1, 1, ... may be considered to be the simplest convergent sequence we can conceive. The process described above consists in comparing, on the average, the terms s n of the sequence 18 The choice of the letters C and H is explained in the two next sub-sections. 59 General remarks on divergent sequences. 465 under consideration with those of the sequence of units: - ' = 1. ' = __ o + Si -f- . . + s n c n - n n w. n j + j + _ ^ + r . This "averaged" comparison of (sj with the unit sequence will be met with again in the case of ihe following processes. The usefulness of this process has already been illustrated above by several examples. We have also seen that it satisfies the two con- ditions 263, I and II, and 111 does not come under consideration at the moment. In 60 and 61 it will further be seen that the con- ditions F (264) are also in wide measure fulfilled. 2. irolder's process, or the H p - process 13 . If with a given sequence (s n ), we proceed from the arithmetic means h n f just formed to their mean and if the sequence (h n ") has a limit in the ordinary sense, lim h n " = $, we say that 14 the sequence s n is limitable Jf 2 with the value s. By 43, 2, every sequence which is limitable H^ (and therefore also every convergent sequence), is also limitable H^, with the same value. The new process therefore satisfies the conditions 263, I, II and III; moreover, its range is wider ihan that of the ^-process, for the series =o for instance, is summable // 2 with the sum .-, but not summable HI nor convergent. In fact, we have here and (h n ') == 1, 0, ~, 0, j! , 0, .... These sequences arc not convergent. On the other hand, the numbers h n " *~T as i s easily calculated. This is precisely the value which one would expect from 1 \ 9 n ' n=0 for x = 1. 13 Holder, O.i Grenzwerte von Reihen an der Konvergenzgrenze. Math. Ann., Vol. 20, pp. 535549. 1882. Here arithmetic means of the kind described are for the first time introduced for a special purpose. 14 The rest of the notation is formed in the same way, // d -lim $ n = s, H % -2a n = s, // 3 (s n )-+.s, etc. but hereafter we shall not mention it specially. 466 Chapter XIII. Divergent series. If the numbers h n " do not tend to a unique limit, we proceed to take their mean /, " I J, " 1 Z, // h '" "*" * ~*~ ' * ' " n Cfi 1 2 * "n w + 1 * ' ' ' " * *' or, in general, for 15 /> ^ 2, the mean ,,.... between the numbers /* ""^ obtained at the previous stage; if these new numbers A^ -> $, for some definite />, we say that the sequence (s n ) is litnitable H v with the value s. It is easy to form sequences which are limitable H 9 for any particular given p, but for no smaller value of p than this 16 . This, together with 43, 2, shows that the /^-processes not only satisfy the conditions 263, I III, but that their range of action is wider for each fixed p ^ 2 than for all smaller values of p. As regards the conditions F, we must again cfer to 60 and 61. 3. Cesaro's process, or the C^-process 17 . We first write n ='S^ Q \ and also, for each k ^ 1, and we now examine the sequence of numbers 18 ,<*- for each fixed k. If, for some value of k, c^ -> 5, we say that the sequence (s n ) is limitable C k with the value s. In the case of the //-process, we cannot obtain simple formulae giving h^ directly in terms of s n , for larger values of p. In the case of the C-process, this is easily done, for we have (*) _ n + k - 1\ , fn + k - 2\ /* - 1\ - > + \ k-i y *i ' U - 1/ Jm 15 Or indeed for p ^ 1, provided we agree to put h 0) = s n and take the H Q - process to be ordinary convergence, as we shall do here and in all analogous cases in future. 16 Write, for instance, (^ 1> ~ 1) ) = 1, 0, 1,0, 1, ... and work backwards to the values of s n . Other examples will be found in the following sections. 17 Cesdro, E.i Sur la multiplication des series. Bull, des sciences math. (2), Vol. 14, pp. 114120. 1890. 18 The denominators of the right hand side are exactly the values of S obtained by starting with the sequence (s n ) == 1, 1, 1, . . . , i. e. they indicate how many of the partial sums s v are comprised in S. Thus the C fc -process again in- volves an "averaged" comparison between a given sequence (s n ) and the unit sequence. 59. General remarks on divergent sequences. 467 or if we wish to go back to the scries -i'a n , with the partial sums s n , This may be proved quite easily by induction, or by noticing that, by 102, n=o n=o so that for every integral k (i-xy~- n =o whence, by 108, the truth of the statement follows 19 . In the following sections we shall enter in detail into this process also, which becomes identical with the preceding one (h n ' = c n ') for ^ 4. AbeVs process, or the A -process. Given a series 2a n with the partial sums s n , we consider the power series If its ladius is ^> 1, and if (for real values of x) the limit lim 2a n x n = lim (1 x) 2s n x n = s a;->l-0 ->l-0 exists, we say that the series 2Ja n is 2Q summable A, and that tlis sequence (s n ) is limit able A 9 with the value s\ in symbols: A-2a n = s, A-\ims n = s. In consequence of Abel's theorem 100, this process also fulfils the permanence condition I, and simple examples show that it fulfils the "extension condition" II; for instance, in the case of the series 2( l) n already used, the limit for x *1 exists. Thus Euler's paiadoxical equation (p. 457) is again justified 19 In view of these last formulae, it is fairly natural to allow non-integral values > 1 for the suffix k also. Such limitation processes of non-integral order were first consistently introduced and investigated by the author (Grenzwerte von Reihen bei der Annaherung an die Konvergenzgrenze, Inaug.-Diss., Berlin 1907). We shall however not enter into this question, either here in the case of the C-process, or later itfi that of the other processes considered. 20 If the product (1 x) 2s n x n is written in the form we see that it is again an "averaged" comparison of the given sequence with the unit sequence which is involved, though in a somewhat different manner. 468 Chapter XIII. Divergent series. by this process. If we now use the more precise form x-^(-i)---J or c^c-iy-A. we thus indicate two perfectly definite processes by which the value - may be obtained from the series J?( l) w . 5. Hitler's process, or the K- process. We saw in 144 that if the first of the two series oo c ^(-l)-. and rt=0 &-- converges, then so does the second, and to the same sum. Simple examples show, however, that the second series may quite well con- verge without the first one doing so: 1. If a n 2= 1, then a Q = 1 and A k a = for &I>1. Accordingly, the two series are 1 1 + 1 H ----- and -L + O-fO + OH ---- the second of which converges to the sum - -. 2. If, for M = 0, 1, 2, ..., a n == 1, 2, 3, 4, ..., then Aa n = - 1, 1, 1, - 1, ..., and for k ^ 2 A*a n = 0, 0, 0, 0, ---- Accordingly, the t\\o series are 1_ 2 + 3-4 -\ ----- and - ~ + + -| ---- . the second of which converges to the sum -^ . 3. Similarly for a n = (n + 1) 3 we find <40 = 7, /I a a =12, -d s a = 6, and, for &>3, zl fe a =0. The two series are thus 1 _ 8 + 27 - 64 + ---- and \ - -J- + ?| - ^ + + + - - -, the second of which converges to the sum -- 5- . o 4. For a n =2 n , A k a Q = ( 1)*. Thus the two series are: ! _ 2 +4 - 8 H ----- and -L - A + 1 - -I H ----- , the second of which converges to the sum y i. e. the sum which we should expect for x = 2 from j- = Z x n . 5. For a n =( l) n z n , d k a = (l + z) k . -The two series are therefore the second of which converges to the sum 1 _ , provided |j?+ l( < 2. 59. General remarks on divergent sequences. 469 If we start with any scries 2a n , without alternately ~f- and signs, the series will be an Eiders transformation of the given series, which we ma> also obtain as follows: The series 2 'a n results from the power series for x = \, hence from for y = . Expanding the latter in powers of y, before substituting y = - , we obtain Eulers transformation. In fact In order to adapt this process for use with any sequence (sj we write, deviating somewhat from the usual notation, o+ a iH ----- Hn-i = s n for w ^!> and So^ ' and also *o'-!-i'H ----- Mn-i = s n ' for w^l, and V^ - It is now easy to verify that 21 for cvvry n .2. We accordingly make the following definition: A sequence (s n ) is said to be limitable E\ with the value s, if the sequence (s n ') just de- fined tends 22 to s. If, without testing the convergence of (s n '), we write 21 From 2a n x n+l = 2 f a n / (2y) n+1 it follows, by multiplication by L-o n-o Hence n=0 whence the relation may at once be inferred. 88 Here also the denominator 2" is obtained from the numerator by replacing: each of the s n 's by 1. Thus we are again concerned with an "averaged" comparison, of a definite kind, between the sequence (s n ) and the unit sequence. 470 Chapter XIII. Divergent series. and in general, for r ^> 1 , ^-TrKD^^ + ffl-^+'-' + C)^" 1 *]; (0,1,2,...). we shall similarly say that the sequence (s n ) is limitdble E r and regard s as its E r - limit, if, for a particular r, s^ >s. Our former theorem 144 (see also 44, 8) then shows in any case that this E- process satisfies the permanence condition I, and the examples given there show that the condition II is also satisfied. This process will be examined further in 63. 6. JRfesz's process, or the Jf^- process 23 . For making the principle of averaged comparison of the sequence (s n ) with the unit sequence more powerful, a principle which, as we saw, lies at the basis of all the former limitation projesscs, a fairly obvious pro cedure consists in attributing arbitrary weights to the various terms s n . If /z , /x 1 , /z 2 , . . . denote any sequence of positive numbers, then _ Mo + Ml + - + J"n is a generalized mean of this kind. In the special case of /^ n -, we speak of a logarithmic mean. As with the //-, C-, or ^-processes, this generalized method of form- ing the mean may of course be repeated, writing, for instance, as in the C-process, >-, and ^=1, and then, for k ^ 1 , n and and then proceeding to investigate, for fixed k ^ 1, the ratio <*) p <*> = "" A tf> for n -> + oo. If these tend to a limit 5, we might say that (s n ) was Hmitable 24 R /Jk with the value s. This definition, however, is not in use. The process in question has reached its great importance only by being transformed into a form more readily amenable to analysis, as 23 Riesz, M. : Sur les series de Dirichlet et les series entieres. Comptes rendus Vol. 149, pp. 909912. J909. 24 Here we add a suffix /* to R k , the notation of the process, as a reference to the sequence (/i n ) used in the formation of the mean. For \JL U = 1, this process reduces exactly to the C^-process. 69. General remarks on divergent sequences. 471 follows: A (complex) function s (t) of the real variable / ^ is defined by s(t) = s v in A^ < t^ A^ (1) (v = 0, 1, 2, . . . ; A^ = 0) with s(0) = 0; then and it is natural to substitute repeated integration for the repeated sum- mation used in the formation of the numbers cr^ and A*\ A A-ple in- tegration a5 gives 000 instead of a n (fc) . Similarly, instead of the numbers A n (ft) , we have to take the values which we obtain by putting s n = 1 in the integrals just written down, i. e. We should then have to deal with the limit (for fixed k) lim A If this limit exists and = s, the sequence (s w ) will be called limitdble -K** with the value s. Here we cannot enter into a more detailed examination of the question whether the two definitions given for the R^- process are really exactly equivalent, or into the elegant and far-reaching appli- cations of the process in the theory of Dirichlets series. (For refer- ences to the literature, see 266.) 7. .Borel's process, or the /J- process. We have just seen how Riesz' process tends to increase the efficiency of the H- or C- pro- cesses, by substituting for the method of averaged comparison be- tween the sequence (s n ) and the unit sequence a more general form of this procedure. The range of Abel's process may be enlarged in a similar way by making use of other series instead of the geometric series there used for purposes of comparison. Taking the exponential series as a particular case, and accordingly considering the quotient of the two series > 4* oo ~n 2s n X -, and 27 J. n=0 n - n-0 ni 25 The equality of the two sides is easily proved by induction, using inte- gration by parts. 472 Chapter XIII. Divergent series, that is to say, the product n=o n wl for x *4-oo> we obtain the process introduced by ZT. Borel 29 . In accordance with it we make the following definition: A sequence (s w ) x n such that the power series ^ s n converges everywhere and the function F(x) just defined tends to a unique limit s as # * -f" > will be called Hinitable B with the value s. In order to illustrate the process to some extent, let us first take 2a n s= 2( l) n once more; then s n = 1 or 0, according as n is even or odd. Accordingly y i-L. + -4-.=^ e *+ e ~* and we have to deal with the limit lim e~*-- - , which is evidently -^ . Thus 2 ( l) n is summable B with the sum -jr . More generally, taking a n = z n > we have, provided only that * 4. -f 1, and which +- - when x -j-oo, provided $l(z) < 1. 77w/s ^ geometric series 2 z n is summable B with the sum = -- throughout the half- plane 27 5ft (-s) < 1. This process also satisfies the permanence condition; for we have If 5 * 5 in the ordinary sense, we can for any given e choose m so "n 28 Sur la sommation des series divergentes, Comptes rendus, Vol 121, p. 1125. 1895, and in many Notes in connection with it. A connected account is given in his Lecons sur les series divergentes, 2 nd ed., Paris 1928. 27 By the C-processes, as shewn in 268, 8, the geometric series is summable, beyond | z \ < 1, only for the boundary points of the unit circle, -f- 1 excepted; by Enters process it is summable throughout the circle | z -f- 1 | < 2, which en- closes the unit circle, with a wide margin; by Borel's process it is summable in the whole half-plane 9ft (z) < 1, the value in this and the preceding cases being every- where . __ . 59. General remarks on divergent sequences. 473 large that | s n s \ < \z for every n > m. The expression on the right hand side is then in absolute value <: r- . J7J i n -t\.*g e~* . J7J *. - | *" + * for positive #'s. Now the product of e~ x and a polynomial of the w th degree tends to when x ~> + oo ; we can therefore choose so large that this product is < s for every x > . For these #'s the whole ex- pression is then < e in absolute value, and our statement is established. 8. The J5 r -process. The range of the process just described is, in a certain sense, extended by substituting other series for 27*-,, in the first instance 27 >--y ( , say, where r is some fixed integer > 1. We accordingly say that a sequence (s n ) is limit able B r with the value s if the quotient of the two functions 00 x rn rn ^i x rn 27 s n 7^-. and 27 , %-:, i. e. the product r e~ x 2 s n 7 -^ n-O ( rn > ! n-o( rw > ! n-0 ( rw > 1 tends to the limit s when # -> -j- oo . (We must, of course, assume again here that the first-named series is everywhere convergent.) Thus the fi-process, for instance, is quite useless for the sequence s n = ( l) n n !, since here s n x ^ = 27 ( l) n x n does not converge for every x\ whereas x rn the series 27 s n ( r, already converges everywhere 28 when we take r 2. 9. Le jRoy's process. We have usually interpreted the limitation processes by saying that by means of them we carry out an "averaged" comparison between the given sequence (s n ) and the unit sequence 1, 1, 1, ... We may look at the matter in a slightly different way. If the numbers s n are the partial sums of the series 27 a n> we have to examine, for instance in the C x -process, the limit of * + *i 4- . . . + s n Here the terms of the series appear multiplied by variable factors which reduce the given series to a finite sum, or at any rate to a series convergent in the old sense. By means of these factors, the influence of distant terms is destroyed or diminished; yet as n increases all the factors tend to 1 and thus ultimately involve all the terms to their full extent. The situation is similar in the case of Abel's process, where we were concerned with the limit of 2a n x n for x -> 1 0; here the effect described above is 28 This does not mean that the # r -process (r > 1) is more favourable than the B-process for every sequence (s n ). On the contrary, there are sequences that are hmitable B but not limitable B a . 474 Chapter XIII. Divergent series. brought about by the factors x n , which, however, increase to 1 as x -> 1 0. This principle appears most clearly as the basis of the following process 29 : The series n=0 is assumed convergent for ^ x < 1. If the function which it defines in that interval tends to a limit 5 as # >1 0, the series Sa n may be called summable R to the value s. This method is not so easily dealt with analytically, and for this reason it is of smaller importance. 10. The most general form of the limitation processes. It will have been noticed that all the processes so far described belong es- sentially to two types: 1. In the case of the first type, from a sequence (s n ), with the help of a matrix (cf. Toeplitz theorem 221) r=(<O a new sequence of numbers *' **o*o + *ii H ----- M ft 5 n + -"> (A = 0, 1, 2,...) is formed by combination of the sequence s , s^j..., s n , ... with the successive rows a k0 , a kl , ..., a fcw , . .. , the assumption being, of course, that the series on the right hand side represents a definite value, i. e. is convergent (in the old sense) 30 . The sequence S Q ', s x ', . . . , s h ', . . . will be called for short the T- transformation 31 of the sequence (s n ) and its n th term, when there is no fear of ambiguity, will be denoted by T (s n ). If the accented sequence (s k f ) is convergent with the limit s, the given sequence is said to be limit able T 'with the value s. In symbols : TMim s n = s or T(s w ) + s . 29 Le Roy: Sur les series divergentes, Annalcs de la Fac. des sciences de Toulouse (2), Vol. 2, p. 317. 1900. * If each row of the matrix T contains only a finite number of terms, this condition is automatically fulfilled. This is the case with the processes 1, 2, 3 and 5. 31 The series 2 aj/ 9 of which the s k ft s are the partial sums, may similarly be called the T- transformation of the series 2 a n with the s n 's as its partial sums. Thus e. g. the series is the Cj- transformation of the series 2 a n . In this sense, all T-processes give more or less remarkable transformations of scries, which may very often he of use in numerical calculations. (This is particularly the case with the E -process). The transformation of the series may equally, of course, be re- garded as the primary process and the transformation of the sequence of partial sums may be deduced from it. Indeed it was in this way that we were led to the E- process. 69. General remarks on divergent sequences. 475 It is at once clear that the processes 1, 2, 3, 5, and the first one described in 6 belong to this type. They differ only in the choice of the matrix T. Theorem 221, 2 also immediately tells us with what matrices we are certain to obtain limitation processes satisfying the permanence condition 32 . 2. In the case of the second type, we deduce from a sequence (s n ) 9 by combining it with a sequence of functions (9w) = 9o (*)> 9i (*)> > 9n (*)> 9 the function pi / M\ fn f*A\ O I / \ | I / \ | where we assume, say, that each of the functions <p n (x) is defined for every x > x and that the series Sy n (x)s n converges for each of these values of x. In that case F (x) is also defined for every x > # , and we may in- vestigate the existence of the limit lim F (x). If the limit exists and = s t x->+v> the sequence (s n ) will be called 33 limit able cp with the value s. By analogy with 221, 2, we shall at once be able to assign conditions under which a process of this type will satisfy the permanence condition. This will certainly be the case if a) for every fixed n, lim 9 n (x) - 0, if b) a constant K. exists such that I <Po(*) I + I <Pi (*) I + . + I <P (*) I < K. for every x > X Q and all rc's, and if c) for x -> + oo Km- It will be noticed that these conditions correspond exactly to the assump- tions 34 a), b) and c) of theorem 221, 2. The proof, which is quite analogous to that of this theorem, may therefore be left to the reader. Borers process evidently belongs to this type, with ep n (x) = e" x ^. The same may be said of Abel's process, if the interval . . . + oo 32 The importance of theorem 221, 2 lies chiefly in the fact that the con- ditions a), b) and c) of the theorem are not merely sufficient, but actually necessary for its general validity. We cannot enter into the question (v. p. 74, footnote 19), but we may observe that in consequence of this fact, the T-processes whose matrix satisfies the conditions mentioned are the only ones which fulfil the permanence condition. 33 In all essentials this is the scheme by means of which O. Perron (Beitrage zur Theorie der divergenten Reihen, Math. Zschr. Vol. 6, pp. 286 310. 1920) classifies all the summation processes. 34 Like these they are not only sufficient, but also necessary for the general validity of the theorem. Further details in H. Raff, Lineare Transformationen bes- chrankter integrierbarer Funktionen, Math. Zeitschr, Vol. 41, pp. 605 629. 1930. 476 Chapter XIII. Divergent series. is projected into the interval ... 1 which is used in the latter, that is, if the series (1 x) 2 s n x n is replaced by the series and the latter is examined for x+-{-oo. In an equally simple manner, it may be seen that Le Roy's process belongs to this type. The second type of limitation process contains the first as a par- ticular case, obtained when x assumes integral values ^ only dp n (k) ~ a kn) u ^ e mere ly use a continuous parameter in the one case, and a discontinuous one in the other. Conversely, in view of 19, def. 4 a, the continuous passage to the limit may be replaced by a discontinuous one, and hence the ^-processes may be exhibited as a sub-class of the T- processes. These remarks, however, are of little use: in further methods of investigation the two types of process nevertheless remain essentially different. It is not our intention to investigate all the processes which come under these two headings from the general points of view indicated above. Let us make only the following remarks. We have already pointed out what conditions the matiixTor sequence of functions (cp n ) must fulfil, in order that the limitation process based on it may satisfy the permanence condition 263, I. Whether the conditions 263, II and III are alo fulfi led, will depend on fuither hypotheses regarding the matrix T or sequence (9^,); this question is accordingly be t left to a separate investigation in each case. The question as to the ex- tent to which the conditions F (264) are fulfilled, cannot be attacked in a general way either, but must be specially examined for each process. One important property alone is con.mon to all the T- and (^-processes, namely their linear character: If two sequences (s n ) and (t n ) are limitable in accordance with one and the same process, the first with the value s, and the second with the value t, then the sequence (as n -{- bt n ), whatever the constants a and b maybe, is also limitable by the same process, with the value as-\-bt. The proof follows immediately from the way in which the process is constructed. Owing to this theorem, all the simplest rules of the algebra of con- vergent sequences (term-by-term addition of a constant, term-by-term multiplication by a constant, term-by term addition or subtraction of two sequences) remain formally unaltered. On the oth^r hand, we must expressly emphasize the fact that the theorem on the influence of a finite number of alterations (42, 7) doe> not necessarily remain valid 35 . 35 For this, the following simple example relating to the B-process was first given by G. H. Hardy: Let s n be defined by the expansion 00 x n sin(e*) = H s n . n-Q n ' $ince e~ x sin (tF) -> as x -> + oo ? the sequences s 0t s lt j ?> . . f is 59. General remarks on divergent sequences. 477 If we wished to give a general and fairly complete survey of the present state of the theory of divergent series, we should now be obliged to enter into a more detailed investigation of the processes which we have described. To begin with, we should have to deal with the questions whether, and to what extent, the individual processes do actually satisfy the stipulations 2G3, II, III and 264; we should have to obtain necessary and sufficient conditions for a series to be summable by a particular process; we should have to find the relations between the ways in which the various processes act, and go further into the questions indicated in No. 10, etc. Owing to lack of space it is of course out of the question to investigate all this in detail. We must be content with examining a few of the processes more particulary; we choose the H-, C-, A-, and E- processes. At the same time we will so arrange the choice of subjects that as far as possible all questions and all methods of proof which play a part in the com- plete theory may at least be indicated. For the rest we must refer to the original papers, of which we may men- tion the following, in addition to those mentioned in the footnotes of this section and of the following sections: 1. The following- give a general survey of the group of problems: Borel, E.: Lecons sur les series divergentes, 2 ld ed., Pans 1928. Bromwich, T. J. PA.- An introduction to the theory of infinite series. London 1908: 2 nd ed. 1926. Hardy. G. H., and S. Chapman A general view of the theory of suramable series. Quarterly Journal Vol. 42, p. 181. 1911. Chapman, S.: On the general theory of summability, with applications to Fourier's and other series. Ibid., Vol. 43, p. 1 1911. Carmichael, R. D.: General aspects of the theory of summable series. Bull, of the American Math. Soc. Vol. 25, pp. 97131. 1919. Knopp t K.: Neuere Untersuchungcn in dcr Theorie dor divergenten Reihen. Jahresber. d. Deutschen Math.-Ver. Vol. 32. pp. 4367. 1923. 2. A more detailed account of the l?^*- process, which is not specially considered in the following sections, is given by Hardy, G. H., and M. Riesz. The general theory of Dirichlet's series. Cambridge 1